In this section, you will:
In 2010, a major earthquake struck Haiti, destroying or damaging over 285,000 homes1. One year later, another, stronger earthquake devastated Honshu, Japan, destroying or damaging over 332,000 buildings,2 like those shown in [link]. Even though both caused substantial damage, the earthquake in 2011 was 100 times stronger than the earthquake in Haiti. How do we know? The magnitudes of earthquakes are measured on a scale known as the Richter Scale. The Haitian earthquake registered a 7.0 on the Richter Scale3 whereas the Japanese earthquake registered a 9.0.4
The Richter Scale is a base-ten logarithmic scale. In other words, an earthquake of magnitude 8 is not twice as great as an earthquake of magnitude 4. It is
times as great! In this lesson, we will investigate the nature of the Richter Scale and the base-ten function upon which it depends.
In order to analyze the magnitude of earthquakes or compare the magnitudes of two different earthquakes, we need to be able to convert between logarithmic and exponential form. For example, suppose the amount of energy released from one earthquake were 500 times greater than the amount of energy released from another. We want to calculate the difference in magnitude. The equation that represents this problem is
where
represents the difference in magnitudes on the Richter Scale. How would we solve for
We have not yet learned a method for solving exponential equations. None of the algebraic tools discussed so far is sufficient to solve
We know that
and
so it is clear that
must be some value between 2 and 3, since
is increasing. We can examine a graph, as in [link], to better estimate the solution.
Estimating from a graph, however, is imprecise. To find an algebraic solution, we must introduce a new function. Observe that the graph in [link] passes the horizontal line test. The exponential function
is one-to-one, so its inverse,
is also a function. As is the case with all inverse functions, we simply interchange
and
and solve for
to find the inverse function. To represent
as a function of
we use a logarithmic function of the form
The base
logarithm of a number is the exponent by which we must raise
to get that number.
We read a logarithmic expression as, “The logarithm with base
of
is equal to
” or, simplified, “log base
of
is
” We can also say, “
raised to the power of
is
” because logs are exponents. For example, the base 2 logarithm of 32 is 5, because 5 is the exponent we must apply to 2 to get 32. Since
we can write
We read this as “log base 2 of 32 is 5.”
We can express the relationship between logarithmic form and its corresponding exponential form as follows:
Note that the base
is always positive.
Because logarithm is a function, it is most correctly written as
using parentheses to denote function evaluation, just as we would with
However, when the input is a single variable or number, it is common to see the parentheses dropped and the expression written without parentheses, as
Note that many calculators require parentheses around the
We can illustrate the notation of logarithms as follows:
Notice that, comparing the logarithm function and the exponential function, the input and the output are switched. This means
and
are inverse functions.
A logarithm base
of a positive number
satisfies the following definition.
For
where,
as, “the logarithm with base
of
” or the “log base
of
is the exponent to which
must be raised to get
Also, since the logarithmic and exponential functions switch the
and
values, the domain and range of the exponential function are interchanged for the logarithmic function. Therefore,
Can we take the logarithm of a negative number?
No. Because the base of an exponential function is always positive, no power of that base can ever be negative. We can never take the logarithm of a negative number. Also, we cannot take the logarithm of zero. Calculators may output a log of a negative number when in complex mode, but the log of a negative number is not a real number.
Given an equation in logarithmic form
convert it to exponential form.
and identify
as
Write the following logarithmic equations in exponential form.
First, identify the values of
Then, write the equation in the form
Here,
Therefore, the equation
is equivalent to
Here,
Therefore, the equation
is equivalent to
Write the following logarithmic equations in exponential form.
is equivalent to
is equivalent to
To convert from exponents to logarithms, we follow the same steps in reverse. We identify the base
exponent
and output
Then we write
Write the following exponential equations in logarithmic form.
First, identify the values of
Then, write the equation in the form
Here,
and
Therefore, the equation
is equivalent to
Here,
and
Therefore, the equation
is equivalent to
Here,
and
Therefore, the equation
is equivalent to
Write the following exponential equations in logarithmic form.
is equivalent to
is equivalent to
is equivalent to
Knowing the squares, cubes, and roots of numbers allows us to evaluate many logarithms mentally. For example, consider
We ask, “To what exponent must
be raised in order to get 8?” Because we already know
it follows that
Now consider solving
and
mentally.
Therefore,
Therefore,
Even some seemingly more complicated logarithms can be evaluated without a calculator. For example, let’s evaluate
mentally.
be raised in order to get
” We know
and
so
Therefore,
**Given a logarithm of the form
evaluate it mentally.**
as a power of
identify
by asking, “To what exponent should
be raised in order to get
”
Solve
without using a calculator.
First we rewrite the logarithm in exponential form:
Next, we ask, “To what exponent must 4 be raised in order to get 64?”
We know
Therefore,
Solve
without using a calculator.
(recalling that
)
Evaluate
without using a calculator.
First we rewrite the logarithm in exponential form:
Next, we ask, “To what exponent must 3 be raised in order to get
”
We know
but what must we do to get the reciprocal,
Recall from working with exponents that
We use this information to write
Therefore,
Evaluate
without using a calculator.
Sometimes we may see a logarithm written without a base. In this case, we assume that the base is 10. In other words, the expression
means
We call a base-10 logarithm a common logarithm. Common logarithms are used to measure the Richter Scale mentioned at the beginning of the section. Scales for measuring the brightness of stars and the pH of acids and bases also use common logarithms.
A common logarithm is a logarithm with base
We write
simply as
The common logarithm of a positive number
satisfies the following definition.
For
We read
as, “the logarithm with base
of
” or “log base 10 of
”
The logarithm
is the exponent to which
must be raised to get
**Given a common logarithm of the form
evaluate it mentally.**
as a power of
to identify
by asking, “To what exponent must
be raised in order to get
”
Evaluate
without using a calculator.
First we rewrite the logarithm in exponential form:
Next, we ask, “To what exponent must
be raised in order to get 1000?” We know
Therefore,
Evaluate
**Given a common logarithm with the form
evaluate it using a calculator.**
followed by [ ) ].
Evaluate
to four decimal places using a calculator.
Rounding to four decimal places,
Note that
and that
Since 321 is between 100 and 1000, we know that
must be between
and
This gives us the following:
Evaluate
to four decimal places using a calculator.
The amount of energy released from one earthquake was 500 times greater than the amount of energy released from another. The equation
represents this situation, where
is the difference in magnitudes on the Richter Scale. To the nearest thousandth, what was the difference in magnitudes?
We begin by rewriting the exponential equation in logarithmic form.
Next we evaluate the logarithm using a calculator:
followed by [ ) ].
The difference in magnitudes was about
The amount of energy released from one earthquake was
times greater than the amount of energy released from another. The equation
represents this situation, where
is the difference in magnitudes on the Richter Scale. To the nearest thousandth, what was the difference in magnitudes?
The difference in magnitudes was about
The most frequently used base for logarithms is
Base
logarithms are important in calculus and some scientific applications; they are called natural logarithms. The base
logarithm,
has its own notation,
Most values of
can be found only using a calculator. The major exception is that, because the logarithm of 1 is always 0 in any base,
For other natural logarithms, we can use the
key that can be found on most scientific calculators. We can also find the natural logarithm of any power of
using the inverse property of logarithms.
A natural logarithm is a logarithm with base
We write
simply as
The natural logarithm of a positive number
satisfies the following definition.
For
We read
as, “the logarithm with base
of
” or “the natural logarithm of
”
The logarithm
is the exponent to which
must be raised to get
Since the functions
and
are inverse functions,
for all
and
for
**Given a natural logarithm with the form
evaluate it using a calculator.**
followed by [ ) ].
Evaluate
to four decimal places using a calculator.
followed by [ ) ].
Rounding to four decimal places,
Evaluate
It is not possible to take the logarithm of a negative number in the set of real numbers.
Access this online resource for additional instruction and practice with logarithms.
Definition of the logarithmic function | For if and only if |
Definition of the common logarithm | Forif and only if |
Definition of the natural logarithm | Forif and only if |
can be evaluated mentally using previous knowledge of powers of
See [link].
can be rewritten as a common logarithm and then evaluated using a calculator. See [link].
What is a base
logarithm? Discuss the meaning by interpreting each part of the equivalent equations
and
for
A logarithm is an exponent. Specifically, it is the exponent to which a base
is raised to produce a given value. In the expressions given, the base
has the same value. The exponent,
in the expression
can also be written as the logarithm,
and the value of
is the result of raising
to the power of
How is the logarithmic function
related to the exponential function
What is the result of composing these two functions?
How can the logarithmic equation
be solved for
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
and then properties of exponents can be applied to solve for
Discuss the meaning of the common logarithm. What is its relationship to a logarithm with base
and how does the notation differ?
Discuss the meaning of the natural logarithm. What is its relationship to a logarithm with base
and how does the notation differ?
The natural logarithm is a special case of the logarithm with base
in that the natural log always has base
Rather than notating the natural logarithm as
the notation used is
For the following exercises, rewrite each equation in exponential form.
For the following exercises, rewrite each equation in logarithmic form.
For the following exercises, solve for
by converting the logarithmic equation to exponential form.
For the following exercises, use the definition of common and natural logarithms to simplify.
For the following exercises, evaluate the base
logarithmic expression without using a calculator.
For the following exercises, evaluate the common logarithmic expression without using a calculator.
For the following exercises, evaluate the natural logarithmic expression without using a calculator.
For the following exercises, evaluate each expression using a calculator. Round to the nearest thousandth.
Is
in the domain of the function
If so, what is the value of the function when
Verify the result.
No, the function has no defined value for
To verify, suppose
is in the domain of the function
Then there is some number
such that
Rewriting as an exponential equation gives:
which is impossible since no such real number
exists. Therefore,
is not the domain of the function
Is
in the range of the function
If so, for what value of
Verify the result.
Is there a number
such that
If so, what is that number? Verify the result.
Yes. Suppose there exists a real number
such that
Rewriting as an exponential equation gives
which is a real number. To verify, let
Then, by definition,
Is the following true:
Verify the result.
Is the following true:
Verify the result.
No;
so
is undefined.
The exposure index
for a 35 millimeter camera is a measurement of the amount of light that hits the film. It is determined by the equation
where
is the “f-stop” setting on the camera, and
is the exposure time in seconds. Suppose the f-stop setting is
and the desired exposure time is
seconds. What will the resulting exposure index be?
Refer to the previous exercise. Suppose the light meter on a camera indicates an
of
and the desired exposure time is 16 seconds. What should the f-stop setting be?
The intensity levels I of two earthquakes measured on a seismograph can be compared by the formula
where
is the magnitude given by the Richter Scale. In August 2009, an earthquake of magnitude 6.1 hit Honshu, Japan. In March 2011, that same region experienced yet another, more devastating earthquake, this time with a magnitude of 9.0.5 How many times greater was the intensity of the 2011 earthquake? Round to the nearest whole number.
is written simply as
must be raised to get
written
must be raised to get
is written as
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