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Evaluate $\text{\hspace{0.17em}}y=\mathrm{ln}\left(500\right)\text{\hspace{0.17em}}$ to four decimal places using a calculator.
Rounding to four decimal places, $\text{\hspace{0.17em}}\mathrm{ln}(500)\approx 6.2146$
Evaluate $\text{\hspace{0.17em}}\mathrm{ln}(\mathrm{-500}).$
It is not possible to take the logarithm of a negative number in the set of real numbers.
Access this online resource for additional instruction and practice with logarithms.
Definition of the logarithmic function | For
$\text{}x0,b0,b\ne 1,$
$y={\mathrm{log}}_{b}\left(x\right)\text{}$ if and only if $\text{}{b}^{y}=x.$ |
Definition of the common logarithm | For $\text{}x0,$ $y=\mathrm{log}\left(x\right)\text{}$ if and only if $\text{}{10}^{y}=x.$ |
Definition of the natural logarithm | For $\text{}x0,$ $y=\mathrm{ln}\left(x\right)\text{}$ if and only if $\text{}{e}^{y}=x.$ |
What is a base $\text{\hspace{0.17em}}b\text{\hspace{0.17em}}$ logarithm? Discuss the meaning by interpreting each part of the equivalent equations $\text{\hspace{0.17em}}{b}^{y}=x\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}{\mathrm{log}}_{b}x=y\text{\hspace{0.17em}}$ for $\text{\hspace{0.17em}}b>0,b\ne 1.$
A logarithm is an exponent. Specifically, it is the exponent to which a base $\text{\hspace{0.17em}}b\text{\hspace{0.17em}}$ is raised to produce a given value. In the expressions given, the base $\text{\hspace{0.17em}}b\text{\hspace{0.17em}}$ has the same value. The exponent, $\text{\hspace{0.17em}}y,$ in the expression $\text{\hspace{0.17em}}{b}^{y}\text{\hspace{0.17em}}$ can also be written as the logarithm, $\text{\hspace{0.17em}}{\mathrm{log}}_{b}x,$ and the value of $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ is the result of raising $\text{\hspace{0.17em}}b\text{\hspace{0.17em}}$ to the power of $\text{\hspace{0.17em}}y.$
How is the logarithmic function $\text{\hspace{0.17em}}f(x)={\mathrm{log}}_{b}x\text{\hspace{0.17em}}$ related to the exponential function $\text{\hspace{0.17em}}g(x)={b}^{x}?\text{\hspace{0.17em}}$ What is the result of composing these two functions?
How can the logarithmic equation $\text{\hspace{0.17em}}{\mathrm{log}}_{b}x=y\text{\hspace{0.17em}}$ be solved for $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ using the properties of exponents?
Since the equation of a logarithm is equivalent to an exponential equation, the logarithm can be converted to the exponential equation $\text{\hspace{0.17em}}{b}^{y}=x,$ and then properties of exponents can be applied to solve for $\text{\hspace{0.17em}}x.$
Discuss the meaning of the common logarithm. What is its relationship to a logarithm with base $\text{\hspace{0.17em}}b,$ and how does the notation differ?
Discuss the meaning of the natural logarithm. What is its relationship to a logarithm with base $\text{\hspace{0.17em}}b,$ and how does the notation differ?
The natural logarithm is a special case of the logarithm with base $\text{\hspace{0.17em}}b\text{\hspace{0.17em}}$ in that the natural log always has base $\text{\hspace{0.17em}}e.\text{\hspace{0.17em}}$ Rather than notating the natural logarithm as $\text{\hspace{0.17em}}{\mathrm{log}}_{e}\left(x\right),$ the notation used is $\text{\hspace{0.17em}}\mathrm{ln}\left(x\right).$
For the following exercises, rewrite each equation in exponential form.
${\text{log}}_{4}(q)=m$
${\mathrm{log}}_{16}\left(y\right)=x$
${\mathrm{log}}_{y}\left(x\right)=\mathrm{-11}$
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