# Math 22 Logarithmic Functions

## Logarithm Function

 The logarithm $\log _{a}x$ is defined as
$\log _{a}x=b$ if and only if $a^{b}=x$ ## Definition of the Natural Logarithmic Function

 The natural logarithmic function, denoted by $\ln x$ , is defined as
$\ln x=b$ if and only if $e^{b}=x$ ## Properties of the Natural Logarithmic Function

 Let $g(x)=\ln x$ 1. The domain of $g(x)$ is $(0,\infty )$ and the range of $g(x)$ is $(-\infty ,\infty )$ 2. The x-intercept of the graph of $g(x)$ is $(1,0)$ 3. The function $g(x)$ is continuous, increasing, and one-to-one.
4. $\lim _{x\to 0^{+}}g(x)=-\infty$ and $\lim _{x\to \infty }g(x)=\infty$ ## Inverse Properties of Logarithms and Exponents

 1.$\ln e^{\sqrt {2}}$ 2.$e^{\ln x}=x$ 3.$\ln {xy}=\ln {x}+\ln {y}$ 4.$\ln {\frac {x}{y}}=\ln x-\ln y$ 5.$\ln {x^{n}}=n\ln x$ Exercises 1 Use the properties of logarithms to rewrite the expression as the logarithm of a single quantity

a) $\ln(x-2)-\ln(x+2)$ Solution:
$\ln(x-2)-\ln(x+2)=\ln {\frac {x-2}{x+2}}$ b) $5\ln(x-6)+{\frac {1}{2}}\ln(5x+1)$ Solution:
$5\ln(x-6)+{\frac {1}{2}}\ln(5x+1)=\ln(x-6)^{5}+\ln[(5x+1)^{\frac {1}{2}}]=\ln[(x-6)^{5}{\sqrt {5x+1}}]$ c) $3\ln x+2\ln y-4\ln z$ Solution:
$\ln x^{3}+\ln y^{2}-\ln z^{4}=\ln {\frac {x^{3}y^{2}}{z^{4}}}$ d) $7\ln(5x+4)-{\frac {3}{2}}\ln(x-9)$ Solution:
$7\ln(5x+4)-{\frac {3}{2}}\ln(x-9)=\ln(5x+4)^{7}-\ln(x-9)^{\frac {3}{2}}=\ln {\frac {(5x+4)^{7}}{(x-9)^{\frac {3}{2}}}}$ Exercises 2 Solve for x.

a) $\ln(2x)=5$ Solution:
$\ln(2x)=5$ , so $e^{5}=2x$ , hence $x={\frac {e^{5}}{2}}$ b) $5\ln x=3$ Solution:
$5\ln x=3$ , so $ln{x^{5}}=3$ , so $e^{3}=x^{5}$ , hence $x={\sqrt[{5}]{e^{3}}}$ 