Logarithmic Functions

Logarithmic Functions

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: $x=b^{y}\,$ 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 $b^{\log _{b}X}=X\,$.

To recap, a logarithm is the inverse function of an exponent.

e.g. The inverse of the function $f(x)=3^{x}$ is $f^{-1}(x)=\log _{3}x$.

In general, $y=b^{x}\iff x=\log _{b}y$, given that 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 > 0} .

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\,}
• 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 \frac{X}{Y} = \log_bX - \log_bY\,}
• 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:

Solving a Logarithmic Equation

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

e.g. Solve 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 x + \log (x+2) = 2} (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} is another way of writing 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_{10}} ).

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 \begin{align} \log x + \log (x+2) &= 2 \\ \log (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{align}}

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