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{{ option.icon }} {{ option.label }} # Evaluating Logarithms

## Logarithm

A logarithm is the inverse of an exponential function. The logarithm of a positive number m is written as and it is read as the logarithm of m with base b.

In the given expressions, b is called the base in both the logarithm and in the exponential expression. Logarithms are defined only for positive values of b and m, where b1. To see the implications of this definition, a particular example will be considered.
In this equation, the definition of logarithm implies that, to get a result of 16, the base 4 would need to be raised to the exponent n.
The following diagram illustrates how a logarithmic form has an equivalent exponential form using a concrete numerical value. ## Common Logarithm

A common logarithm is a logarithm of base 10. For example, is called the common logarithm of 1000. It is equal to 3 because 103 is 1000. Since common logarithms are used so often, the base does not need to be written.

In the above identity, m is a positive number. Recalling the definition of a logarithm, the common logarithm of m can be defined for positive values of m.

## The Natural Logarithm

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

Although it is correct to write the natural logarithm is more commonly written as

## Rewrite the logarithmic and exponential expressions

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For the following expressions, rewrite them in either logarithmic or exponential form.
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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,
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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Evaluate the following logarithms.
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When evaluating logarithms, it can be helpful to think about what the expression means. asks the exponent to which 3 must be raised to equal 81. Since, 3333=81,
The second expression, 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 As it turns out, negative exponents yield fractions.
It can be helpful to rewrite 1000 as a power of 10. Since 1000=103,
Thus,