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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 x ( 64 ) = y

x y = 64

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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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