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<H2>What is the logarithm</H2> <H4>Definition of logarithm</H4> <BLOCKQUOTE>The <B>logarithm</B> of a number <B>x</B> to a given base <B>b</B> is the power <B>y</B> to which the base must be raised to get the number <B>x</B>. In mathematical symbols, {{{y = log(b,x)}}}. An expression {{{log(b,x)}}} is read "the logarithm, base b, of x".</BLOCKQUOTE> In expression {{{log(b,x)}}} the real number <B>x</B> is called <B>an argument</B>, while the real number <B>b</B> is called <B>the base</B>. The logarithm {{{log(b,x)}}} is defined for all positive real numbers <B>x</B>. So, any positive real number <B>x</B> can be an argument. The base of the logarithm can be any positive real number, except of 1. <B>Examples</B> 1) {{{log(2,8) = 3}}}. An argument of the logarithm is equal to '8' here, the base is equal to '2'. The value of {{{log(2,8)}}} is equal to 3 because {{{2^3=8}}}. 2) {{{log(10,100) = 2}}}. An argument of the logarithm is equal to '100' in this example, the base is equal to '10'. The value of {{{log(10,100)}}} is equal to 2 because {{{10^2=100}}}. It follows the logarithm definition, that if {{{y = log(b,x)}}}, then {{{x = b^y}}}. Why? Simply, if {{{y = log(b,x)}}}, then the base value <B>b</B> raised to the degree <B>y</B> should be equal to <B>x</B>. This is exactly what the expression {{{x = b^y}}} states. Actually, both these expressions, {{{y = log(b,x)}}} and {{{x = b^y}}}, are equivalent. The direct consequence of it is an equality {{{x = b^log(b,x)}}}, as well as an equality {{{y = log(b,(b^y))}}}. <B>Examples for the last two formulas</B> 3) {{{8 = 2^log(2,8)}}}; 4) {{{100 = 10^log(10,100)}}}. <B>More examples of logarithms</B> 5) {{{log(2,(1/8)) = -3}}}. An argument of the logarithm is equal to {{{1/8}}} here, the base is equal to 2. The value of {{{log(2,(1/8))}}} is equal to -3 because {{{2^(-3)=1/8}}}. 6) {{{log(10,0.01) = -2}}}. An argument of the logarithm is equal to 0.01 in this example, the base is equal to 10. The value of {{{log(10,0.01)}}} is equal to -2 because {{{10^(-2)=0.01}}}. You see that the logarithms themselves can be negative. <B>More examples of logarithms with the base value less than 1</B> 7) {{{log((1/2),8) = -3}}}. Here an argument of the logarithm is equal to 8, the base is equal to {{{1/2}}}. The value of {{{log((1/2),8)}}} is equal to -3 because {{{(1/2)^(-3)=8}}}. 8) {{{log((1/2),(1/8)) = 3}}}. In this example an argument of the logarithm is equal to {{{1/8}}}, the base is equal to {{{1/2}}}. The value of {{{log((1/2),1/8)}}} is equal to 3 because {{{(1/2)^3=1/8}}}. <H4>Logarithmic function</H4> Let us consider the logarithmic function {{{y=log(b,x)}}} for the case when the base <B>b</B> of the logarithm is greater than 1, for example, for the base value <B>b=2</B>. For values of x = 1/4, 1/2, 1, 2, 4, 8 the corresponding values of logarithm {{{log(2,x)}}} are equal to -2, -1, 0, 1, 2, 3. The plot of the logarithmic function {{{y=log(2,x)}}} is shown in <B>Figure 1</B> below. First, you see that for this case values of the logarithmic function are negative for values of <B>x</B> less than 1 and positive for values of <B>x</B> greater than 1. You see also that the logarithmic function {{{y=log(2,x)}}} is monotonically increased when the argument <B>x</B> is increased. This is the typical plot and the typical behavior of the logarithmic function for the case when the base is greater than 1. <TABLE cellspacing="40"> <TD> {{{ graph( 320, 200, -2, 10, -4, 5, log(2, x ) ) }}} Figure 1. Logarithmic function {{{y=log(2,x)}}} </TD> <TD> {{{ graph( 320, 200, -2, 10, -4, 5, log(0.5, x ) ) }}} Figure 2. Logarithmic function {{{y=log(0.5,x)}}} </TD> </TABLE> Let us consider the logarithmic function {{{y=log(b,x)}}} for the case when the base <B>b</B> of the logarithm is less than 1, for example, for the base value <B>b=1/2=0.5</B>. For values of x = 1/4, 1/2, 1, 2, 4, 8 the corresponding values of logarithm {{{log(0.5,x)}}} are equal to 2, 1, 0, -1, -2, -3. The plot of the logarithmic function {{{y=log(0.5,x)}}} is shown in <B>Figure 2</B> above. In contrast to the previous example, you see that for this case values of the logarithmic function are positive for values of <B>x</B> less than 1 and negative for values of <B>x</B> greater than 1. You see also that the logarithmic function {{{y=log(0.5,x)}}} is monotonically decreased when the argument <B>x</B> is increased. This is the typical plot and the typical behavior of the logarithmic function for the case when the base is less than 1. My other lessons in this site on logarithms, logarithmic equations and relevant word problems are - <A HREF=http://www.algebra.com/algebra/homework/logarithm/Properties-of-the-logarithm.lesson>Properties of 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.
