Subsection 7.3.1 A Logarithm is an Exponent
Suppose that a colony of bacteria doubles in size every day. If the colony starts with 50 bacteria, how long will it be before there are 800 bacteria?
We answer questions of this type by writing and solving an exponential equation. The initial value of the population is and the growth factor is Thus, the function
gives the number of bacteria present on day and we must solve the equation
We are looking for an unknown exponent on base 2. Dividing both sides by 50 yields
This equation asks the question: "To what power must we raise 2 in order to get 16?"
Because we see that the solution of the equation is You can check that solves the original problem:
The unknown exponent that solves the equation is called the base logarithm of The exponent in this case is and we write this fact as
In other words, we solved an exponential equation by computing a logarithm. We make the following definition.
Definition 7.3.1. Definition of Logarithm.
the
base logarithm of is the exponent to which
must be raised in order to yield
We write the logarithm as
Some logarithms, like some square roots, are easy to evaluate, while others require a calculator. We’ll start with the easy ones.
Example 7.3.2.
Here are some examples of logarithms.
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We see that in each example the logarithm is the exponent we need on the given base.
Checkpoint 7.3.4. Practice 1.
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(What exponent on 3 gives me 81?)
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(What exponent on 10 gives me
)
From the definition of a logarithm and the examples above, we see that the following two statements are equivalent.
Logarithms and Exponents: Conversion Equations.
This equivalence tells us that the logarithm,
is the same as the
exponent in
We see again that
a logarithm is an exponent; it is the exponent to which
must be raised to yield
For example, to convert the equation
to exponential form, we note that the base is
and the logarithm is
so the exponent on base 5 will be 3, like this:
The
conversion equations allow us to convert from logarithmic to exponential form, or vice versa. It will help you to become familiar with the conversion, because we will use it frequently.
As special cases of the equivalence above, we can compute the following useful logarithms.
Some Useful Logarithms.
Example 7.3.5.
Here are some more examples. Once again, we are looking for the exponent on the given base
to get
Checkpoint 7.3.6. QuickCheck 1.
Subsection 7.3.2 Logs and Exponential Equations
We use logarithms to solve exponential equations, just as we use square roots to solve quadratic equations. Consider the two equations
We solve the first equation by taking a square root, and we solve the second equation by computing a logarithm:
The operation of taking a base logarithm is the inverse operation for raising the base to a power, just as extracting square roots is the inverse of squaring a number.
Every exponential equation can be rewritten in logarithmic form by using the conversion equations. Thus,
are equivalent statements, just as
are equivalent statements. Rewriting an equation in logarithmic form is a basic strategy for finding its solution.
Example 7.3.7.
Checkpoint 7.3.8. Practice 2.
Rewrite each equation in logarithmic form.
We can solve an exponential equation by converting the equation to logarithmic form.
Example 7.3.9.
Solve each equation by converting to logarithmic form.
Solution.
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In this case we can see by inspection that the solution is
or
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The log form is
which is not easy to evaluate without a calculator.
Checkpoint 7.3.10. Practice 3.
Solve each equation by converting to logarithmic form.
Checkpoint 7.3.11. QuickCheck 2.
Answer the question or complete the statement.
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What is the inverse operation for squaring a number?
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What is the inverse operation for raising to a power?
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Every exponential equation can be rewritten in logarithmic form by using the
.
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A logarithm is an unknown
.