Evaluating Logarithms

Concept

Logarithm

A logarithm is the inverse of an exponential function. The logarithm of a positive number m can be written as follows. The expression $lo g_{b} (m)$ is read as $lo g$ base b of m and states that raising b to the $n^{th}$ power yields m.

Here, b is both the base of the logarithm and the base of the exponent. For example, the value of n in

lo g_{4} (16) = n

is given by the exponent to which 4 would be raised to result in 16.

The logarithmic form and exponential form are equivalent.

The relationship between base and exponent for logarithms and powers

Concept

Common Logarithm

A common logarithm is a logarithm of base 10. For example, $lo g_{10} (1000)$ is equal to 3 because 103 is equal to 1000.

Since $lo g_{10}$ is used so often, it is sometimes written without a base. For positive values of m, the common $lo g$ of m can be defined as follows.

Concept

The Natural Logarithm

A natural logarithm is a logarithm with base e. This means that $ln (m)$ equals the exponent to which e must be raised to equal m.

Although it is correct to write $lo g_{e},$ the natural logarithm is more commonly written as $ln .$

Rewrite the logarithmic and exponential expressions

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Exercise

For the following expressions, rewrite them in either logarithmic or exponential form.

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Solution

We will rewrite each expression one at a time using the relationship

Notice that the first and third are written as logarithms and the second and fourth are written as exponents. Since a base is not written on the first logarithm, we know it's base 10. Furthermore, 2 is the exponent. Thus,

The same reasoning applies for the third expression,

lo g_{2} (32) = 5 .

Using the same relationship in the opposite way, we can rewrite the second and fourth expression. In 82=64, it can be seen that 8 is the base and 2 is the exponent. This means that 64 is the value of which we take the logarithm.

For the last expression, this yields

To summarize, the following expressions are equivalent.

Evaluate the logarithms

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Exercise

Evaluate the following logarithms.

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Solution

When evaluating logarithms, it can be helpful to think about what the expression means.

lo g_{3} (81)

asks the exponent to which 3 must be raised to equal 81. Since, 3⋅3⋅3⋅3=81,

The second expression,

ln (e),

is a logarithm of base e. It asks the exponent to which e must be raised to equal e. Since e1=e,

Notice that, unlike the other expressions, the last, which is of base 10, contains a fraction. Thus, we must consider the exponent to which 10 is raised to equal

\frac{1}{1 0 0 0} .

As it turns out, negative exponents yield fractions.

It can be helpful to rewrite 1000 as a power of 10. Since 1000=103,

Thus,

lo g (\frac{1}{1 0 0 0}) = - 3 .

Evaluating Logarithms

Concept

Logarithm

Concept

Common Logarithm

Concept

The Natural Logarithm

Example

Rewrite the logarithmic and exponential expressions

Example

Evaluate the logarithms