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. 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 \lim_{x\to 0^+} g(x)=-\infty}
 and 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 \lim_{x\to\infty} g(x)=\infty}

Inverse Properties of Logarithms and Exponents

 1.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 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}

 

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