6.3 Logarithmic functions (Page 5/9)

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Evaluating a natural logarithm using a calculator

Evaluate $y = \ln (500)$ to four decimal places using a calculator.

Press [LN] .
Enter $500,$ followed by [ ) ] .
Press [ENTER] .

Rounding to four decimal places, $\ln (500) \approx 6.2146$

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Try It

Evaluate $\ln (−500) .$

It is not possible to take the logarithm of a negative number in the set of real numbers.

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Media

Access this online resource for additional instruction and practice with logarithms.

Introduction to Logarithms

Key equations

Definition of the logarithmic function	For $x > 0, b > 0, b \neq 1,$ $y = \log_{b} (x)$ if and only if $b^{y} = x .$
Definition of the common logarithm	For $x > 0,$ $y = \log (x)$ if and only if $10^{y} = x .$
Definition of the natural logarithm	For $x > 0,$ $y = \ln (x)$ if and only if $e^{y} = x .$

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 $b$ can be evaluated mentally using previous knowledge of powers of $b .$ See [link] and [link] .
Common logarithms can be evaluated mentally using previous knowledge of powers of $10.$ See [link] .
When common logarithms cannot be evaluated mentally, a calculator can be used. See [link] .
Real-world exponential problems with base $10$ can be rewritten as a common logarithm and then evaluated using a calculator. See [link] .
Natural logarithms can be evaluated using a calculator [link] .

Section exercises

Verbal

What is a base $b$ logarithm? Discuss the meaning by interpreting each part of the equivalent equations $b^{y} = x$ and $\log_{b} x = y$ for $b > 0, b \neq 1.$

A logarithm is an exponent. Specifically, it is the exponent to which a base $b$ is raised to produce a given value. In the expressions given, the base $b$ has the same value. The exponent, $y,$ in the expression $b^{y}$ can also be written as the logarithm, $\log_{b} x,$ and the value of $x$ is the result of raising $b$ to the power of $y .$

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How is the logarithmic function $f (x) = \log_{b} x$ related to the exponential function $g (x) = b^{x} ?$ What is the result of composing these two functions?

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How can the logarithmic equation $\log_{b} x = y$ be solved for $x$ 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 $b^{y} = x,$ and then properties of exponents can be applied to solve for $x .$

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Discuss the meaning of the common logarithm. What is its relationship to a logarithm with base $b,$ and how does the notation differ?

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Discuss the meaning of the natural logarithm. What is its relationship to a logarithm with base $b,$ and how does the notation differ?

The natural logarithm is a special case of the logarithm with base $b$ in that the natural log always has base $e .$ Rather than notating the natural logarithm as $\log_{e} (x),$ the notation used is $\ln (x) .$

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Algebraic

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

${log}_{4} (q) = m$

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${log}_{a} (b) = c$

$a^{c} = b$

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$\log_{16} (y) = x$

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$\log_{x} (64) = y$

$x^{y} = 64$

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$\log_{y} (x) = −11$

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Questions & Answers

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Source: OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6

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