# 4.3 Logarithmic functions  (Page 5/9)

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## Evaluating a natural logarithm using a calculator

Evaluate $\text{\hspace{0.17em}}y=\mathrm{ln}\left(500\right)\text{\hspace{0.17em}}$ to four decimal places using a calculator.

• Press [LN] .
• Enter $\text{\hspace{0.17em}}500,$ followed by [ ) ] .
• Press [ENTER] .

Rounding to four decimal places, $\text{\hspace{0.17em}}\mathrm{ln}\left(500\right)\approx 6.2146$

Evaluate $\text{\hspace{0.17em}}\mathrm{ln}\left(-500\right).$

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.

## Key equations

 Definition of the logarithmic function For if and only if Definition of the common logarithm For if and only if Definition of the natural logarithm For if and only if

## Key concepts

• The inverse of an exponential function is a logarithmic function, and the inverse of a logarithmic function is an exponential function.
• Logarithmic equations can be written in an equivalent exponential form, using the definition of a logarithm. See [link] .
• Exponential equations can be written in their equivalent logarithmic form using the definition of a logarithm See [link] .
• Logarithmic functions with base $\text{\hspace{0.17em}}b\text{\hspace{0.17em}}$ can be evaluated mentally using previous knowledge of powers of $\text{\hspace{0.17em}}b.\text{\hspace{0.17em}}$ See [link] and [link] .
• Common logarithms can be evaluated mentally using previous knowledge of powers of $\text{\hspace{0.17em}}10.\text{\hspace{0.17em}}$ See [link] .
• When common logarithms cannot be evaluated mentally, a calculator can be used. See [link] .
• Real-world exponential problems with base $\text{\hspace{0.17em}}10\text{\hspace{0.17em}}$ can be rewritten as a common logarithm and then evaluated using a calculator. See [link] .
• Natural logarithms can be evaluated using a calculator [link] .

## Verbal

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 $\text{\hspace{0.17em}}f\left(x\right)={\mathrm{log}}_{b}x\text{\hspace{0.17em}}$ related to the exponential function $\text{\hspace{0.17em}}g\left(x\right)={b}^{x}?\text{\hspace{0.17em}}$ 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 $\text{\hspace{0.17em}}\mathrm{ln}\left(x\right).$

## Algebraic

For the following exercises, rewrite each equation in exponential form.

${\text{log}}_{4}\left(q\right)=m$

${\text{log}}_{a}\left(b\right)=c$

${a}^{c}=b$

${\mathrm{log}}_{16}\left(y\right)=x$

${\mathrm{log}}_{x}\left(64\right)=y$

${x}^{y}=64$

${\mathrm{log}}_{y}\left(x\right)=-11$

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