Logarithmic Functions

Logarithmic Functions

In mathematics you can find the inverse of an exponential function by switching x and y around: $y=b^{x}\,$ becomes $x=b^{y}\,$. 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: $x=b^{y}\,$ becomes $y=\log _{b}x\,$. If a log is given without a written b then b=10. Also with logarithmic functions, b > 0 and $b\neq 1$. There are 2 cases where the log is equal to x: $\log _{b}b^{X}=X\,$ and $b^{\log _{b}X}=X\,$.

Laws of Logarithmic Functions

When X and Y are positive.

• $\log _{b}XY=\log _{b}X+\log _{b}Y\,$
• $\log _{b}{\frac {X}{Y}}=\log _{b}X-\log _{b}Y\,$
• 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_b X^k = k \log_bX\,}

Change of Base

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

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_2 8 = \frac { \log 8}{ \log 2} = \frac {.9}{.3} = 3\,} now check 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 2^3 = 8\,}

Resources

• Logarithmic Functions, Book Chapter
• Guided Notes