log a b = x if and only if a x = b. Use the properties of logarithms that you derived in Explorations 1–3 to evaluate each logarithmic expression. The notation is read “the logarithm (or log) base of .” The definition of a logarithm indicates that a logarithm … Use the rule below when transforming log equation into an exponential equation. These are often known as logarithmic properties, which are documented in the table below. Logarithms can be used to make calculations easier. 1 Log Formula | Logarithm Rules Practice | Logarithm Tutorial | Exercise – 2. These seven (7) log rules are useful in expanding logarithms, condensing logarithms, and solving logarithmic equations. In addition, we discuss how to evaluate some basic logarithms including the use of the change of base formula. 1.1.1 First Law of the logarithms – ( logarithm addition rule ); 1.1.2 Second Law of the logarithms – ( logarithm subtract rule ); 1.1.3 Third Law of the logarithms – ; 1.1.4 Base Change Rules . However, I have intentionally left one out to discuss it here in detail. Examples on Rn and Rm×n ... (similar proof as for log-sum-exp) Convex functions 3–10. log a xy = log a x + log a y. By the inverse of the Fundamental Theorem of Calculus, since lnx is de ned as an integral, it is di erentiable and its derivative is the integrand 1=x. Principle Properties of Logarithm. The slide rule below is presented in a disassembled state to facilitate cutting. Video transcript. Use respectively the changes of variable u = −log(t) and u2 = −log(t) in (1). We use “if and only if” or the double-headed arrow, ⟷, to denote a biconditional statement. We will also discuss the common logarithm, log(x), and the natural logarithm, ln(x). In this video, I prove the power, product and quotient rule for logarithms. That is, copy the common base then add the exponents. Expressed mathematically, x is the logarithm of n to the base b if b x = n, in which case one writes x = log b n.For example, 2 3 = 8; therefore, 3 is the logarithm of 8 to base 2, or 3 = log 2 8. We assume that M(log(N)) is O(1) group operations (this is true for all the algebraic groups of interest in this book). Proof for the Quotient Rule 5 0 obj In fact, the useful result of 10 3 = 1000 1024 = 2 10 can be readily seen as 10 log 10 2 3.. Proof that log a M N = log a M log a N: Examples 3 (a) log 2 40 log 2 5 = log 2 40 5 = log 2 8: If x= log 2 8 then 2x= 8 = 23;so x= 3: (b) If log 3 5 = 1:465 then we can nd log 3 0 6: Since 3=5 = 0 36;then log 3 0 6 = log 3 5 = log 3 3 log 3 5: Now log 3 3 = 1;so that log … Key Point log a x m = mlog a x 7. ComputingallΦlei i (g) andΦlei i Many logarithmic expressions may be rewritten, either expanded or condensed, using the three properties above. The complex logarithm, exponential and power functions In these notes, we examine the logarithm, exponential and power functions, where the arguments∗ of these functions can be complex numbers. Step 1: Let {\color{red}m }= {\log _b}x and {\color{blue}n} = {\log _b}y. x��\K��qV�"|X�ht���i��!Z��E($��7�pP�bA���_֣+��fv� Proof of Logarithmic Rules Rule 1: The Power Rule log a x n = n log ax Proof: Let m = log ax. Rule for write Mantissa and Characteristic: To make the mantissa positive ( In case the value of the logarithm of a number is negative), subtract 1 from the integral part and add to the decimal part.. For example log 10 (0.5) = – 0.3010 Step 1: Assume that {\color{red}m }= {\log _b}x and {\color{blue}n} = {\log _b}y. Step 5: In step 1, we suppose \large{{\color{red}m} = {\log _b}x}. Review : Common Graphs – This section isn’t much. \large{k = {\log _a}x \,\,\to\,\, x = {a^k}}. We will also discuss the common logarithm, log(x), and the natural logarithm, ln(x). \large{{\log _b}\left( x \right) = {\log _b}\left( {{a^{\large{{\color{red}k}}}}} \right)}. \Large{{\LARGE{{x \over y}}} = {b^{\,m - n}}}. $1 per month helps!! The properties on the right are restatements of the general properties for the natural logarithm. Expanding is breaking down a complicated expression into simpler components. So, let's just review real quick what a logarithm even is. Proof for the Product Rule. If you see In particular, we are interested in how their properties differ from the properties of the corresponding real-valued functions.† 1. So I decided to prove 2 of them myself. Then the following properties of exponents hold, provided that all of the expressions appearing in a particular equation are de ned. LOGARITHMS AND THEIR PROPERTIES Definition of a logarithm: If and is a constant , then if and only if . It is true that a logarithmic equation can be expressed as an exponential equation, and vice versa. The choice of base \color{green}\large{b} is intentional because we want to get rid of the base \large{b} on the right side of the equation. 7182818284 59 ... ). 1.1 Logarithm formula sheet ( Laws of Logarithms ) . Welcome to this presentation on logarithm properties. Step 3: We want to prove the Quotient Rule of Logarithm so we will divide x by y, therefore our set-up is \Large{x \over y}. Step 4: Now, apply the Power Rule of Logarithm on the right side of the exponential equation to bring down the exponent k. Then solve for k by dividing both sides of the equation by {\log _b}\left( a \right). In other words, logarithms and exponentials are equivalent. (Note that f (x)=x2 is NOT an exponential function.) The Matrix Logarithm: from Theory to Computation Nick Higham School of Mathematics ... University of Edinburgh, March 2014. In the same fashion, since 10 2 = 100, then 2 = log 10 100. Proof: Let log b x = p such that b p = x … (i), and. Proof: Step 1: Let m = log a x and n = log a y. Outline 1 Properties, Formulas & Applications 2 Computation Nick Higham Matrix Logarithm 2 / 33. (2) This result can be proved directly from the definition of the matrix exponential given by eq. \large{{\log _b}\left( {xy} \right) = {\log _b}x + {\log _b}y}, \large{{\log _b}\left( {x \cdot y} \right) = {\log _b}x + {\log _b}y}, \large{{\log _b}\left( {\Large{{{x \over y}}}} \right) = {\log _b}x - {\log _b}y}. A Proof of the Logarithm Properties. Common Logarithm: The logarithm with base 10 is called the Common Logarithm and is denoted by omitting the base. Most of the time, we are just told to remember or memorize these logarithmic properties because they are useful. ln x is called the natural logarithm and is used to represent log e x , where the irrational number e 2 : 71828. Proof of Property (1) For inverse functions, In addition, since the inverse of a logarithmic function is an exponential function, I would also … Logarithm Rules Read More » The derivative of the natural logarithm function is the reciprocal function. In this section we will introduce logarithm functions. \large{xy = \left( {{b^m}} \right)\left( {{b^n}} \right)}. Hence, we may assume that the factorisation of N is known. As every di erentiable function is continuous, therefore lnx is continuous. This section is always covered in my class. Let's do some work on logarithm properties. \large{{\log _b}\left( {xy} \right) = {\log _b}\left( {{b^{m + n}}} \right)}, \large{{\log _b}\left( {xy} \right) = m + n}. Hello. Definition. We give the basic properties and graphs of logarithm functions. The logarithm of x raised to the power of y is y times the logarithm of x. log b (x y) = y ∙ log b (x) For example: log 10 (2 8) = 8∙ log 10 (2) Derivative of natural logarithm. This is the second law. Step 4: Take log a of both sides and evaluate log a xy = log a a m+n log a xy = (m + n) log a a log a xy = m + n log a xy = log a x + log a y. We just need to clean up the right side of the equation by putting the variable k in front of the log expression. The number is the exponent to which a must be raised to obtain M. That is, (1) The logarithm to the base a of a raised to a power equals that power. ( a m) n = a mn 3. Proof: One can factor N using trial division in O(BM(log(N))) bit operations, where M(n) is the cost of multiplying n-bit integers. Logarithm, the exponent or power to which a base must be raised to yield a given number. I Integrals involving logarithms. Study the proofs of the logarithm properties: the product rule, the quotient rule, and the power rule. q.e.d. %�쏢 (b) The integral of y = x nis Z x dx = x(n+1) (n +1), for n 6= −1. From this we can readily verify such properties as: log 10 = log 2 + log 5 and log 4 = 2 log 2. Study the proofs of the logarithm properties: the product rule, the quotient rule, and the power rule. In particular, we are interested in how their properties differ from the properties of the corresponding real-valued functions.† 1. This formula allows you to 2. Loading... Save for later. Some important properties of logarithms are given here. In the equation is referred to as the logarithm, is the base , and is the argument. Express quantity in Exponential form The above three properties are the important one for logarithms. Step 3: Since we are proving the product property, we will multiply x by y. Simplify by applying the product rule of exponent. {\log _b}\left({ \Large{{{x \over y}}}} \right) = {\log _b}x - {\log _b}y, \large{\log _b}\left( {{x^k}} \right) = k \cdot {\log _b}x. Property 1: If A, B ≡ AB − BA = 0, then e A+ B= e eB = e eA. \large{{\log _b}\left( {{x^k}} \right) = mk}, \large{{\log _b}\left( {{x^k}} \right) = \left( {{{\log }_b}x} \right)k}. Condensing is the reverse of this process. 1. Natural Logarithms (Sect. About this resource. First, the following properties are easy to prove. @ of the log of any base. If a and b are positive numbers and r is a rational number, we have the following properties: I (i) ln1 = 0This follows from our previous discussion on the graph of y = ln(x). Preview. The change of base formula for logarithms. Natural Logarithm: The logarithm with base e is called the Natural Logarithm and is denoted by ‘ln’. The logarithmic properties listed above hold for all bases of logs. The Root Formula is a special case of the Power Rule and therefore doesn't require the separate proof. When. log b y = q such that b q = y … (ii) Well, this just means that x to the n equals a. I think we already know that. Next, we have the inverse property. Thanks to all of you who support me on Patreon. 7.2) I Definition as an integral. Change of Base: find the calculator value Created: Jan 7, 2014. Simplify it using the Logarithm of a Base to a Power Rule. Proof of the logarithm quotient and power rules. PROPERTIES OF LOGARITHMIC FUNCTIONS EXPONENTIAL FUNCTIONS An exponential function is a function of the form f (x)=bx, where b > 0 and x is any real number. Express powers as factors. where m and n are integers in properties 7 and 9. Properties of Exponents and Logarithms Exponents Let a and b be real numbers and m and n be integers. Step 2: Express each logarithmic equation as an exponential equation. math class free. The power law property is actually derived by the power rule of exponents and relation between exponent and logarithmic operations. Let m and n be arbitrary positive numbers, be any real numbers, then. This section usually gets a quick review in my class. Change of Base Formula or Rule. 1. a ma n= a + 2. Since the conditional statement and its converse are both true, they are a biconditional statement. (0�����B�H-�M{5PorOc���j_f�eeC]S"�˴$��F�Ii��M�'%I_�3j2�\3�(,bc���Y�y"%,!v}��*5\��ϘPz�GI� A�_+��D�˟/�d�*��g��Y�c^���bU���M�^6���ua$>S��mj�1���� ���!e�����a|����j�=r��~F�Ѱ�Q��k�& ���#�h$��{�V�9O�b����R�r~�܀���J����7',������(�_΋��&x��"��Q��ꬆ��x�����nꙐ-@�g0t!ɮĤddb�i�;~^Q��Ȕbd���{��BDۅc6��g[��`(�(�9�@���b�W�!�q�(�]���4h��/�MC1�0�+ f��;*�~��&pM�`�a��t����J�0�HLQ�w�>�=�[�|p�_a]�W��! Common Logarithms of Numbers N 0 1 2 34 56 7 8 9 10 0000 0043 0086 0128 0170 0212 0253 0294 0334 0374 11 0414 0453 0492 0531 0569 0607 0645 0682 0719 0755 This results to the Quotient Property of Logarithm as intended. That is, (2) The proof uses the fact that and are inverses. Thus we have by = x. logb1=0logbb=1logb1=0logbb=1 For example, log51=0log51=0 since 50=150=1 and log55=1log55=1 since 51=551=5. Use the power property to simplify log 3 9 4. log 3 9 4 = 4 log 3 9 You could find 9 4, but that wouldn’t make it easier to simplify the logarithm. Step 3: Take the logarithms with a different base of both sides of the exponential equation, x = {a^k}. In this section we will introduce logarithm functions. Change of Base: find the calculator value Logarithm, the exponent or power to which a base must be raised to yield a given number. Logarithm product rule. We established it in Step 1. extractor, is used in [13] for de ning the soundness property of interactive proofs of knowledge. This means essentially that there is an equivalence between logarithmic statements and exponential statements. The first three operations below assume that x = b c and/or y = b d, so that log b (x) = c and log b (y) = d. Logarithm of a Product 1. Well, the left side is now simply M n (since a log a M is M) — and the right side simplifies too, because a log a M n is simply M n.(a raised to a power and logarithm base a are opposite operations).But this still wasn't a "textbook" polished proof, because I was using a question mark instead of equal sign to mark that I don't yet know if the two things are equal. Proof. !Tи�+�9-RU8?����|�xJ0��Z���'���g߱���dzH��&��j�������NJ{0�7k(�u���)��F0���&�C����G��?R��f����S��D����m �u����Zj���BR�'���NY���ש�1�O�hy����\)t�GX��= ���&�؆%��s�Z The log rule is called the Change-of-Base Formula.. LOGARITHMIC FUNCTIONS log b x =y means that x =by where x >0, b >0, b ≠1 Think: Raise b to the power of y to obtain x. y is the exponent. Natural Logarithm FunctionGraph of Natural LogarithmAlgebraic Properties of ln(x) LimitsExtending the antiderivative of 1=x Di erentiation and integrationLogarithmic di erentiationsummaries De nition and properties of ln(x). In the properties given next,M and a are positive real numbers, with and r is any real number. Step 2: Write in exponent form x = a m and y = a n. Step 3: Multiply x and y x • y = a m • a n = a m+n. Natural Logarithm: The logarithm with base e is called the Natural Logarithm and is denoted by ‘ln’. The slide rule below is presented in a disassembled state to facilitate cutting. Prove the Four (4) Properties of Logarithms, 1) Product Property: {\log _b}\left( {{x \cdot y}} \right) = {\log _b}x + {\log _b}y, 2) Quotient Property: {\log _b}\left( {\Large{{{x \over y}}}} \right) = {\log _b}x - {\log _b}y, 3) Power Property: {\log _b}\left( {{x^k}} \right) = k \cdot {\log _b}x, 4) The Change of Base Property: {\log _a}x = {\Large{{{{{\log }_b}x} \over {{{\log }_b}a}}}}. Here is an alternate proof of the quotient rule for logarithms using the fact that a reciprocal is a negative power: ... difference, or product of logarithms with the same base, write an equivalent expression as a single logarithm. These two properties, ln1 = 0 and d dx lnx = 1 x, characterize the logarithm. Here, Characteristic = 1 & Mantissa = 0.3979 Note: Mantissa is always written as positive number. Step 1: Suppose \large{{\color{red}m} = {\log _b}x}. simplify the natural logarithm of products and quotients. Notes on the Matrix Exponential and Logarithm HowardE.Haber Santa Cruz Institute for Particle Physics University of California, Santa Cruz, CA 95064, USA May 6, 2019 Abstract In these notes, we summarize some of the most important properties of the matrix exponential and the matrix logarithm. Proofs of Logarithm Properties or Rules The logarithm properties or rules are derived using the laws of exponents. Justifying the logarithm properties. The logarithm of number b on the base a (log a b) is defined as an exponent, in which it is necessary raise number a to gain number b (The logarithm exists only at positive numbers). D`'z������Y���JLr%�_����ك�����L�.�~8���U������9n+)�h�Z�? 81 = {3^4} \,\,\to\,\, {\log _3}81 = 4, 2. M N N M log b =log b −log b log 8 1 7 56 log 8 56 log 8 7 log 8 = 8 = Log a (m n) = log a m + log a n. In general, log a (x 1, x 2, x 3,…, x n) = log a x 1 This section usually gets a quick review in my class. Recall that the logarithmic and exponential functions “undo” each other. Definition as an integral Recall: (a) The derivative of y = xn is y0 = nx(n−1), for n integer. ( ab ) m= a b 4. a m a n = a m n, a 6= 0 5. a b m = a m b m Use the power property to rewrite log 3 9 4 as 4log 3 9. • basic properties and examples • operations that preserve convexity • the conjugate function ... • logarithm: logx on R++ Convex functions 3–3. I placed the rule below for your convenience. Common Logarithm: The logarithm with base 10 is called the Common Logarithm and is denoted by omitting the base. The converse of the original statement is true as well by definition, that is, “if x = {b^y}, then {\log _b}x=y.” We can also write it symbolically as: \Large{\color{blue}x} = {b^{\color{red}y}} \,\,\to \,\,{\log _b}{\color{blue}x} = {\color{red}y}, 1. Nearly all of the results of these notes are well Contents. We de ne a new function lnx = Z x 1 1 t dt; x > 0: This function is called the natural logarithm. Logarithmic Laws and Properties. ;�J\��[��o����ٷ�/7[3���]?�h5�h���FL>xe��f#'mE��.Tt�'����7.���b.��f����R�v���4�ģߝ��Lk?� ��/ �7���TʚJ��kz��O!6j� The logarithm of a multiplication of x and y is the sum of logarithm of x and logarithm of y. log b (x ∙ y) = log b (x) + log b (y). 31) ln 10 31) A) a - b B) ab C) ln a + ln b D) a + b 32) ln 20 32) A) 2a + b B) 2a + 2b C) 4b D) a + b Write as the sum and/or difference of logarithms. stream 1.1.4.1 Some other Properties of logarithms Here is a set of practice problems to accompany the Solving Logarithm Equations section of the Exponential and Logarithm Functions chapter of the notes for Paul Dawkins Algebra course at Lamar University. (1). The complex logarithm, exponential and power functions In these notes, we examine the logarithm, exponential and power functions, where the arguments∗ of these functions can be complex numbers. Use properties of logarithms to write each logarithm in terms of a and b. I was reading a school algebra book about logarithm function (on $\mathbb{R}^+$).There were several properties without proof. A 3. Theorem 4. f (x) = ln(x) The derivative of f(x) is: f ' … The value of logarithmic terms like $\log_{b}{(m^{\displaystyle n})}$ can be calculated by power law identity of logarithms. Remarks: log x always refers to log base 10, i.e., log x = log 10 x . In fact, the useful result of 10 3 = 1000 1024 = 2 10 can be readily seen as 10 log 10 2 3.. That’s the reason why we are going to use the exponent rules to prove the logarithm properties below. Natural Logarithm FunctionGraph of Natural LogarithmAlgebraic Properties of ln(x) LimitsExtending the antiderivative of 1=x Di erentiation and integrationLogarithmic di erentiationExponentials De nition and properties of ln(x). {\Large{{1 \over 8}}} = {2^{ - 3}} \,\,\to\,\, {\log _2}\left( {\Large{{{1 \over 8}}}} \right) = - 3, 3. Couple examples below illustrate how to use logarithm properties together. Step 2: Express {\color{red}k} = {\log _a}x as an exponential equation. Apply the power property first. A) log 24 B) 30 C) 5 D) 100,000 Suppose that ln 2 = a and ln 5 = b. It has importance in growth and decay problems. 6.2 Properties of Logarithms 439 log 2 8 x = log 2(8) log 2(x) Quotient Rule = 3 log 2(x) Since 23 = 8 = log 2(x) + 3 2.In the expression log 0:1 10x2, we have a power (the x2) and a product.In order to use the Product Rule, the entire quantity inside the logarithm must be raised to the same exponent. You may be able to recognize by now that since 3 2 = 9, log 3 9 = 2. \Large{\log _b}{\color{blue}x} = {\color{red}y} \,\,\to \,\,{\color{blue}x} = {b^{\color{red}y}}, 1. Then, using the de nition of logarithms, we can rewrite this as m = log ax )x = am Now, x = am xn = (am)n Writing back in logarithmic form and substituting, we have log ax n = nm log ax n = n log ax Rule 2: The Product Rule log axy = log This conditional statement is true by definition. By elementary changes of variables this historical definition takes the more usual forms : Theorem 2 For x>0 Γ(x)=0 tx−1e−tdt, (2) or sometimes Γ(x)=20 t2x−1e−t2dt. Rules or Laws of Logarithms In this lesson, you’ll be presented with the common rules of logarithms, also known as the “log rules”. Condensing is the reverse of this process. If you're seeing this message, it means we're having trouble loading external resources on our website. �{��Z�Y�~-賈A(5����������~Xi����,t�r�/���كG]�~u����7+y����ᗟ�ϣ?�~s����W��a&m��n�y���,�8�5*Y��a���b�k7U��������P��Bz�~ٞ�o�d���C�Nư~K#�I�/ePl�����~�o������$��^�����(]'�ǘ#����O���$��M�,�ӔWm���P��/'X#�Iu�E!U�rO>�&e|��ƻ�E��_��z5�Ƨ� ?n�!�lfr�@w��3, *�3j}�Fe�X�rv /yC Section 3 The Natural Logarithm and Exponential The natural logarithm is often written as ln which you may have noticed on your calculator. Expanding is breaking down a complicated expression into simpler components. Step 5: Our last step is to substitute back the expression for k = {\log _a}x. log 3 9 4 = 8. In other words, logarithms are exponents. Three Laws of logarithm proof and proof of change of base formula is explained in this video.For any query please comment or email at query.mdc@gmail.com. In addition, we discuss how to evaluate some basic logarithms including the use of the change of base formula. Step 2: Rewrite \large{{\color{red}m} = {\log _b}x} as an exponential equation. The choice of the base doesn’t matter as long as the base is greater than zero but doesn’t equal 1. These are true for either base. {\log _2}32 = 5 \,\,\to\,\, 32 = {2^5}, 2. First, consider the conditional statement “if {\log _b}x = y, then x = {b^y}.” We can also write the statement symbolically to denote implication using the rightward arrow, →. Many logarithmic expressions may be rewritten, either expanded or condensed, using the three properties above. In the same manner, if we have an exponential statement, we can transform it into a logarithmic statement. The power law property is actually derived by the power rule of exponents and relation between exponent and logarithmic operations. \large{m = {\log _b}x \,\,\to x\,\, = {b^m}}. Properties: 1. Answer. 16 = {64^{\small{{{2 \over 3}}}}} \,\,\to\,\, {\log _{64}}16 = {\Large{{2 \over 3}}}. logb(bx)=xblogbx=x,x>0logb(bx)=xblogbx=x,x>0 For example, to evaluate log(100)log(100), we can re… Logarithm Properties Cheat Sheet ... 2018 - properties of logarithms for a typical proof of these laws change of base formula for logarithms let a and b be positive numbers that are not 3 / 14. The product rule can be used for fast multiplication calculation using addition operation. Author: Created by PatrickJMT. You da real mvps! The logarithm of the product of two numbers say x, and y is equal to the sum of the logarithm of the two numbers. Read more. Review : Exponential and Logarithm Equations – How to solve exponential and logarithm equations. PROPERTIES OF LOGARITHMS EXAMPLES 1. log b MN =log b M +log b N log 50 +log 2 =log 100 =2 Think: Multiply two numbers with the same base, add the exponents. (3) Proof. In this video from PatricJMT we look at proof of the logarithm properties. Logarithm of a Quotient 1. How do I trust God with certainty when my situation seems so uncertain? Metallica Fade to … So if I write, let's say I write log base x of a is equal to, I don't know, make up a letter, n. What does this mean? First, we consider some elementary properties. Formulas and properties of logarithms. This section is always covered in my class. here because the logarithmic function is only defined for a domain of positive x. a. log 4 163 b. log 3 81 −3 c. ln e2 d. + ln e5 2 ln e6 − ln e5 e. log 5 75 − log 5 3 f. log 4 2 + log 4 32 CONSTRUCTING VIABLE ARGUMENTS To be profi cient in math, you need to understand Always remember this handy rule which is the \large{{\log _b}b = 1}. Proof: Immediate from JCF formula and scalar case. Logarithm power rule. Proof that log a MN= log a M+ log a N: Examples 2 (a) log 6 4 + log 6 9 = log 6 (4 9) = log 6 36: If x= log 6 36;then 6x= 36 = 62: Thus log 6 4 + log 6 9 = 2: (b) log 5 20 + log 4 1 4 = log 5 20 1 4: Now 20 1 4 = 5 so log 5 20 + log 4 1 4 = log 5 5 = 1: Quiz. Step 3: Raise both sides of the equation to the \large{k} power. The above property defines that logarithm of a positive number m to the power n is equal to the product of n and log of m. Example: log 2 10 3 = 3 log 2 10. … Proofs of Logarithm Properties Read More » Review : Common Graphs – This section isn’t much. Focus your attention on the right side of the equation. Proof of the Logarithm Properties (no rating) 0 customer reviews. These are true for either base. Niall Horan - This Town (Lyric Video) Strangers, again. From this we can readily verify such properties as: log 10 = log 2 + log 5 and log 4 = 2 log 2. log b x = log a x log a b To do so, we let y = log b x and apply these as exponents on the base b: by = blog b x By log property (I) of page 87, the right side of this equation is sim-ply x.