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.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 e^{\ln x}=x}

 
 3.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 \ln{xy}=\ln{x}+\ln{y}}

 
 4.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 \ln{\frac{x}{y}}=\ln x - \ln y}

 
 5.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 \ln{x^n}=n\ln x}

Exercises Use the properties of logarithms to rewrite the expression as the logarithm of a single quantity

a) 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 \ln(x-2)-\ln(x+2)}

Solution:
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 \ln(x-2)-\ln(x+2)=\ln \frac{x-2}{x+2}}

b) $5\ln(x-6)+{\frac {1}{2}}\ln(5x+1)$

Solution:
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 5\ln{x-6}+\frac{1}{2}\ln(5x+1)=\ln(x-6)^5+\ln[(5x+1)^{\frac{1}{2}}]}

c) 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 3\ln x+2\ln y -4\ln z}

Solution:
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 f'(x)=6xe^{3x^2}}

d) 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 7\ln (5x+4)-\frac{3}{2}\ln (x-9)}

Solution:
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 f'(x)=6xe^{3x^2}}

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