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4.3 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 ) ≈ 6.2146

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

Key equations

Definition of the logarithmic function For     x > 0 , b > 0 , b ≠ 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 ≠ 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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Source:  OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
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