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 $x={\sqrt[{5}]{e^{3}}}$

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