03Blogarithms

toc =Logarithms =

In the equation y = a x , x is the index (or power or exponent), The index, x, is also called a ** logarithm **.

y = a x is sometimes called an ** indicial equation ** (or ** exponential equation **) because the variable is in the index (or exponent).

math y=a^x \text{ can also be written as } \log_a \, (y) =x \qquad. math (provided a > 0)

For example: math 8 = 2^3 \text{ can also be written as } \log_2 \, (8) = 3 \qquad. math {this means that with a base of 2, the index to give an answer of 8 is 3}

For example, math \log_{10} \, (73) = 1.863 \text{ is telling us that } 10^{1.863} = 73 \qquad. math


 * NOTE: ** log a (y) is only defined when y > 0

The equivalence between the two equations is shown using the double headed arrow: math 10^4 = 10,000 \; \iff \; \log_{10}\,(10,000) = 4 \qquad. math

{The double headed arrow is notation for **iff**, which means "if and only if"}

Logarithm Laws
math \log_a \,(mn) = \log_a \,(m) + \log_a \, (n) \qquad. math

math \log_a \, \big(\frac{m}{n}\big) = \log_a\, (m) - \log_a\, (n) \qquad. math

math \log_a \, \big( m^p \big) = p \, \log_a\, (m) \qquad. math

math \log_a \, (1) = 0 math

math \log_a \, \big(\frac{1}{m}\big) = - \log_a \, (m) \qquad. math

math \log_a\, (a) = 1 math

It is often useful to change the base of a logarithmic expression.

** Change-of-base rule: **
math \log_a\,(y) = \dfrac{\log_b\,(y)}{\log_b\,(a)} \qquad. math

math \log_2 \, (x) = \dfrac{\log_{10} \, (x)}{\log_{10}\,(2)} \qquad. math
 * For example **

** Example 1 **
Write the following in index form: math \\ a) \;\; \log_{10}(1000)=3 \qquad . \\ . \\ b) \; \; \log_x (81)=4 \qquad. \\ . \\ c) \; \; \log_5 (x)=4 math

__** Solution: **__ math a) \quad \log_{10}(1000)=3 \; \; \iff \; \;10^3=1000 \qquad . math

math b) \quad \log_x(81)=4 \;\; \iff \;\; x^4=81 \qquad . math

math c)\quad \log_5 (x)=4 \;\; \iff \;\; 5^4=x \qquad . math

** Example 2 **
math \\ \text{Using } a^x=y \;\; \iff \;\; \log_a (y)=x \text{ simplify the following:} \qquad. \\ . \\ a) \;\; \log_4 (64) \\ . \\ b) \;\; \log_3\big(\frac{1}{27}\big) math

__** Solution: **__

math \\ a) \;\; \text{Let } \log_4 (64)=x \qquad .\\ . \\ \text{then } \log_4 (64)=x \;\; \iff \;\; 4^x=64 \qquad . math math \\ . \qquad 4^x=4^3 \qquad \qquad . \\ . \\ . \qquad x=3 math

NOTE: We could have also simplifed this by using log laws: math \\ a) \quad \log_4 (64) \qquad . \\ . \\ . \qquad = \log_{4} \big( 4^3 \big) \qquad . \\ . \\ . \qquad = 3\log_{4} (4) \qquad .\\ . \\ . \qquad = 3 math

math b) \;\; \text{Let } \log_3\big(\frac{1}{27}\big)=x \qquad . math math \text{then } \log_3\big(\frac{1}{27}\big) = x \;\; \iff \;\; 3^x = \frac{1}{27} \qquad . math math 3^x = \dfrac{1}{3^3} \qquad . math math 3^x = 3^{-3} \qquad . math

math x =-3 \qquad. math

** Example 3 **
math \text{Simplify }\log_4 (6) - \log_4 (96) \qquad. math

__** Solution: **__

math \log_4(6) - \log_4 (96) \qquad. math

math . \qquad= \log_4\Bigg( \dfrac{6}{96} \Bigg) \qquad. math

math .\qquad= \log_4 \Bigg( \dfrac{1}{16} \Bigg) \qquad. math

math .\qquad=\log_4\Bigg(\dfrac{1}{4^2}\Bigg) \qquad. math

math .\qquad=\log_4 \Big( 4^{-2} \Big) \qquad. math

math .\qquad=-2\log_4 (4) \qquad. math

math .\qquad= -2 \qquad. math

** Example 4 **
math \text{Simplify } \dfrac{\log_2\,(8)}{log_2\, (64)} \qquad. math

__** Solution: **__ math . \quad \dfrac{\log_2 \,(8)}{\log_2 \, (64)} \qquad. math

math .\qquad= \dfrac{\log_2 \, \big(2^3 \big) }{\log_2 \, \big( 2^6 \big) } \qquad. math

math .\qquad= \dfrac{3\log_2 \,(2)}{6\log_2 (2) } \qquad. math

math .\qquad=\dfrac{3}{6} \qquad. math

math .\qquad=\dfrac{1}{2} \qquad. math

** Example 5 **
Express the following in the form Alog n (B) math 5\log_3 (x) + 3- \log_3 \big(x^2 \big) \qquad \qquad. math

__** Solution: **__

math 5\log_3 (x)+3-\log_3 \big( x^2 \big) \qquad. math

math .\qquad=\log_3 \big( x^5 \big) + 3\log_3 (3) - \log_3 \big(x^2 \big) \qquad. math

math .\qquad= \log_3 \big(x^5 \big) + \log_3 \big(3^3 \big) - \log_3 \big( x^2 \big) \qquad. math

math .\qquad = \log_3 \Bigg( \dfrac{x^5 \times 3^3}{x^2} \Bigg) \qquad. math

math .\qquad = \log_3 \Big( (3x)^3 \Big) \qquad. math

math . \qquad = 3\log_3 (3x) \qquad. math

For another site that explains this idea, go here: MathsIsFun

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