Lesson Properties of the logarithm
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<H2>Properties of the Logarithm</H2> The definition of the logarithm is given in lesson <A HREF=http://www.algebra.com/algebra/homework/logarithm/what-is-the-logarithm.lesson>WHAT IS the logarithm</A> in this site. Now we consider properties of logarithms - formulas for the logarithm of a product, logarithm of a quotient, logarithm of a power and logarithm of a root. <H3>Product Rule</H3> <BLOCKQUOTE>The logarithm of a product of two positive real numbers is equal to the sum of logarithms of factors: {{{log(b,(MN)) = log(b,M) + log(b,N)}}} </BLOCKQUOTE> <B>Examples</B> 1) According to the Product Rule, {{{log(2,(4*8)) = log(2,4) + log(2,8) = 2 + 3 = 5}}}. Check it by making the direct calculation: {{{log(2,(4*8)) = log(2,32) = 5}}}. You get exactly the same number as the Product Rule produces! 2) According to the Product Rule, {{{log(10,(0.1*1000)) = log(10,0.1) + log(10,1000) = -1 + 3 = 2}}}. Verify it by making the direct calculation: {{{log(10,(0.1*1000)) = log(10,100) = 2}}}. You get the same result as the Product Rule produces. <B>Proof of the Product Rule</B> Let {{{log(b,M)=x}}}, {{{log(b,N)=y}}}. Then {{{b^x=M}}}, {{{b^y=N}}} due to the logarithm definition (see lesson <A HREF=http://www.algebra.com/algebra/homework/logarithm/what-is-the-logarithm.lesson>WHAT IS the logarithm</A>). Multiplying these two equations, you obtain {{{MN=b^x*b^y = b^(x+y)}}}. This means {{{log(b,(MN))= x+y}}} due to the logarithm definition. Substitute {{{x=log(b,M)}}} and {{{y=log(b,N)}}} to the last formula, and you obtain the required formula {{{log(b,(MN)) = log(b,M) + log(b,N)}}}. <H3>Quotient Rule</H3> <BLOCKQUOTE>The logarithm of a quotient of two positive real numbers is equal to the logarithm of the dividend minus the logarithm of the divisor: {{{log(b,(M/N)) = log(b,M) - log(b,N)}}} </BLOCKQUOTE> <B>Examples</B> 3) According to the Quotient Rule, {{{log(2,(8/2)) = log(2,8) - log(2,2) = 3 - 1 = 2}}}. Check it: {{{log(2,(8/2)) = log(2,4) = 2}}}. You get exactly the same number as the Quotient Rule produces. 4) According to the Quotient Rule, {{{log(10,(1000/10)) = log(10,1000) - log(10,10) = 3 - 1 = 2}}}. Verify it: {{{log(10,(1000/10)) = log(10,100) = 2}}}. You get the same result as the Quotient Rule produces. <B>Proof of the Quotient Rule</B> Let {{{log(b,M)=x}}}, {{{log(b,N)=y}}}. Then {{{b^x=M}}}, {{{b^y=N}}} due to the logarithm definition (see lesson <A HREF=http://www.algebra.com/algebra/homework/logarithm/what-is-the-logarithm.lesson>WHAT IS the logarithm</A>). Dividing these two equations, you obtain {{{M/N=b^x/b^y = b^(x-y)}}}. This means {{{log(b,(M/N))= x-y}}} due to the logarithm definition. Substitute {{{x=log(b,M)}}} and {{{y=log(b,N)}}} to the last formula, and you obtain the required formula {{{log(b,(M/N)) = log(b,M) - log(b,N)}}}. <H3>Power Rule</H3> <BLOCKQUOTE>The logarithm of a power of positive number is equal to the exponent times the logarithm of the number: {{{log(b,(M^p)) = p*log(b,M)}}} </BLOCKQUOTE> <B>Examples</B> 5) According to the Power Rule for logarithms, {{{log(2,(8^2)) = 2*log(2,8)= 2*3 = 6}}}. Check it: {{{log(2,(8^2)) = log(2,64) = 6}}}. You get the same number as the Power Rule produces. 6) According to the Power Rule, {{{log(10,(1000^3)) = 3*log(10,1000) = 3*3 = 9}}}. Check it: {{{log(10,(1000^3)) = log(10,1000000000) = 9}}}. You get the same result as the Power Rule produces. <B>Proof of the Power Rule</B> Let {{{log(b,M)=x}}}. Then {{{b^x=M}}} due to the logarithm definition (see lesson <A HREF=http://www.algebra.com/algebra/homework/logarithm/what-is-the-logarithm.lesson>WHAT IS the logarithm</A>). Raising both sides to the p-th power, you obtain {{{M^p=(b^x)^p = b^(p*x)}}}. This means {{{log(b,(M^p))= p*x}}} due to the logarithm definition. Substitute {{{x=log(b,M)}}} to the last formula, and you obtain the required formula {{{log(b,(M^p)) = p*log(b,M)}}}. <H3>Logarithm of a Root</H3> <BLOCKQUOTE>The logarithm of a root of positive number is equal to the logarithm of the number divided by the index of the root: {{{log(b,root(m,p)) = (1/m)*log(b,p)}}} </BLOCKQUOTE> <B>Examples</B> 7) According to the Root Formula for logarithms, {{{log(2,sqrt(3)) = (1/2)*log(2,3)}}}. Compare it with what the Power Rule produces: {{{log(2,sqrt(3)) = log(2,(3^(1/2))) = (1/2)*log(2,3)}}}. The Root Formula produces exactly the same number as the Power Rule. 8) According to the Root Formula, {{{log(10,sqrt(2)) = (1/2)*log(10,2)}}}. Compare it with what the Power Rule produces: {{{log(10,sqrt(2)) = log(10,(2^(1/2))) = (1/2)*log(10,2)}}}. The Root Formula produces exactly the same result as the Power Rule. The Root Formula is a special case of the Power Rule and therefore doesn't require the separate proof. <B>Couple examples below illustrate how to use logarithm properties together.</B> 9) Calculate {{{log(b,((x^3)/(yz)))}}}. {{{log(b,((x^3)/(yz))) = log(b,(x^3))-log(b,(yz)) = 3*log(b,x)-(log(b,y)+log(b,z)))}}}. 10) Calculate {{{log(b,root(4,(x*y)/(z^3)))}}}. {{{log(b,root(4,(x*y)/(z^3))) = (1/4)*log(b,((x*y)/(z^3))) = (1/4)*(log(b,x)+log(b,y)-3*log(b,z))}}}. <H3>Summary</H3><BLOCKQUOTE>{{{log(b,(MN)) = log(b,M) + log(b,N)}}} {{{log(b,(M/N)) = log(b,M) - log(b,N)}}} {{{log(b,(M^p)) = p*log(b,M)}}} {{{log(b,root(m,p)) = (1/m)*log(b,p)}}}</BLOCKQUOTE> My other lessons in this site on logarithms, logarithmic equations and relevant word problems are - <A HREF=http://www.algebra.com/algebra/homework/logarithm/what-is-the-logarithm.lesson>WHAT IS the logarithm</A> - <A HREF=http://www.algebra.com/algebra/homework/logarithm/change-of-base-formula-for-logarithms.lesson>Change of Base Formula for logarithms</A> - <A HREF=https://www.algebra.com/algebra/homework/logarithm/Evaluate-logarithms-without-using-a-calculator.lesson>Evaluate logarithms without using a calculator</A> - <A HREF=https://www.algebra.com/algebra/homework/logarithm/Simplifying-expressions-with-logarithms.lesson>Simplifying expressions with logarithms</A> - <A HREF=http://www.algebra.com/algebra/homework/logarithm/How-to-solve-logarithmic-equations.lesson>Solving logarithmic equations</A> - <A HREF=https://www.algebra.com/algebra/homework/logarithm/Solving-advanced-logarithmi%D1%81-equations.lesson>Solving advanced logarithmic equations</A> - <A HREF=https://www.algebra.com/algebra/homework/logarithm/Solving-really-interesting-and-educative-problem-on-logarithmic-equation-containing-a-HUGE-underwater-stone.lesson>Solving really interesting and educative problem on logarithmic equation containing a HUGE underwater stone</A> - <A HREF=https://www.algebra.com/algebra/homework/logarithm/Proving-equalities-with-logarithms.lesson>Proving equalities with logarithms</A> - <A HREF=https://www.algebra.com/algebra/homework/logarithm/Solving-logarithmic-inequalities.lesson>Solving logarithmic inequalities</A> - <A HREF=http://www.algebra.com/algebra/homework/logarithm/Using-logarithms-to-solve-real-world-problems.lesson>Using logarithms to solve real world problems</A> - <A HREF=https://www.algebra.com/algebra/homework/logarithm/Solving-problem-on-Newton-Law-of-cooling.lesson>Solving problem on Newton Law of cooling</A> - <A HREF=https://www.algebra.com/algebra/homework/logarithm/Population-growth-problems.lesson>Population growth problems</A> - <A HREF=https://www.algebra.com/algebra/homework/logarithm/Radioactive-decay-problems.lesson>Radioactive decay problems</A> - <A HREF=https://www.algebra.com/algebra/homework/logarithm/Carbon-dating-problems.lesson>Carbon dating problems</A> - <A HREF=https://www.algebra.com/algebra/homework/logarithm/Bacteria-growth-problems.lesson>Bacteria growth problems</A> - <A HREF=https://www.algebra.com/algebra/homework/logarithm/A-medication-decay-in-a-human%27s-body.lesson>A medication decay in a human's body</A> - <A HREF=https://www.algebra.com/algebra/homework/logarithm/Problems-on-appreciated-depreciated-values.lesson>Problems on appreciated/depreciated values</A> - <A HREF=https://www.algebra.com/algebra/homework/logarithm/Inflation-and-Salary-problems.lesson>Inflation and Salary problems</A> - <A HREF=https://www.algebra.com/algebra/homework/logarithm/Miscellaneous-problems-on-exponential-growth-decay.lesson>Miscellaneous problems on exponential growth/decay</A> - <A HREF=https://www.algebra.com/algebra/homework/logarithm/Problems-on-discretely-compound-accounts.lesson>Problems on discretely compound accounts</A> - <A HREF=https://www.algebra.com/algebra/homework/logarithm/Problems-on-continuously-compound-accounts.lesson>Problems on continuously compound accounts</A> - <A HREF=https://www.algebra.com/algebra/homework/logarithm/Tricky-problem-on-solving-a-logarithmic-system-of-equations.lesson>Tricky problem on solving a logarithmic system of equations</A> - <A HREF=https://www.algebra.com/algebra/homework/logarithm/Uninterrupted-withdrawing-money-from-a-retirement-fund.lesson>Entertainment problem: Uninterrupted withdrawing money from a retirement fund</A> - <A HREF=https://www.algebra.com/algebra/homework/logarithm/Entertainment-problems-on-logarithms.lesson>Entertainment problems on logarithms</A> - <A HREF=https://www.algebra.com/algebra/homework/logarithm/Entertainment-problems-on-exponential-growth.lesson>Entertainment problems on exponential growth</A> - <A HREF=https://www.algebra.com/algebra/homework/logarithm/Upper-level-problem-on-solving-logarithmic-equations.lesson>Upper level problems on solving logarithmic equations</A> - <A HREF=http://www.algebra.com/algebra/homework/logarithm/OVERVIEW-of-lessons-on-logarithms-logarithmic-eqns-and-relevant-word-probs.lesson>OVERVIEW of lessons on logarithms, logarithmic equations and relevant word problems</A> Use this file/link <A HREF=https://www.algebra.com/algebra/homework/quadratic/lessons/ALGEBRA-I-YOUR-ONLINE-TEXTBOOK.lesson>ALGEBRA-I - YOUR ONLINE TEXTBOOK</A> to navigate over all topics and lessons of the online textbook ALGEBRA-I.
