Logarithmic Functions

Logarithmic Functions

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In mathematics you can find the inverse of an exponential function by switching x and y around: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y = b^x\,} becomes Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x = b^y\,} . The problem arises on how to find the value of y. The logarithmic function solved this problem. All conversions of logarithmic function into an exponential function follow the same pattern: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x = b^y\,} becomes Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y = \log_b x\,} . If a log is given without a written b then b=10. Also with logarithmic functions, b > 0 and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b \ne 1} . There are 2 cases where the log is equal to x: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \log_bb^X = X\,} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b^{\log_bX} = X\,} .

Laws of Logarithmic Functions

When X and Y are positive.

• Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \log_bXY = \log_bX + \log_bY\,}
• $\log _{b}{\frac {X}{Y}}=\log _{b}X-\log _{b}Y\,$
• $\log _{b}X^{k}=k\log _{b}X\,$

Change of Base

When x and b are positive real numbers and are not equal to 1. Then you can write $\log _{a}x\,$ as ${\frac {\log _{b}x}{\log _{b}a}}$. This works for the natural log as well. here is an example:

Solving a Logarithmic Equation

A logarithmic equation is an equation wherein one or more of the terms is a logarithm.

e.g. Solve $\lg x+\lg(x+2)=2$

${\begin{aligned}\lg x+\lg(x+2)&=2\\\lg(x(x+2))&=2\\x(x+2)&=100\\x^{2}+2x&=100\\(x+1)^{2}&=101\\x+1&={\sqrt {101}}\\x&=-1\pm {\sqrt {101}}\end{aligned}}$

$\log _{2}8={\frac {\log 8}{\log 2}}={\frac {.9}{.3}}=3\,$ now check $2^{3}=8\,$

Resources

• Logarithmic Functions, Book Chapter
• Guided Notes