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Exponential and Logarithmic Functions

Evaluating Logarithms

Concept

Logarithm

A logarithm is the inverse of an exponential function. The logarithm of a positive number can be written as follows. The expression is read as base of and states that raising to the power yields

Here, is both the base of the logarithm and the base of the exponent. For example, the value of in is given by the exponent to which would would be raised to result in 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 For example, is equal to because is equal to

the Connection between the base and the exponent for common logarithms

Since is used so often, it is sometimes written without a base. For positive values of the common of can be defined as follows.

Concept

The Natural Logarithm

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

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

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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 Furthermore, 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 it can be seen that is the base and is the exponent. This means that is the value of which we take the logarithm. For the last expression, this yields To summarize, the following expressions are equivalent.

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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. asks the exponent to which must be raised to equal Since, The second expression, is a logarithm of base It asks the exponent to which must be raised to equal Since Notice that, unlike the other expressions, the last, which is of base contains a fraction. Thus, we must consider the exponent to which is raised to equal As it turns out, negative exponents yield fractions. It can be helpful to rewrite as a power of Since Thus,

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