Exponential functions and logarithmic functions are closely tied. In fact, they are so closely tied we could say a logarithm is actually an exponent in disguise. A logarithm, or log, is just another way to write an exponential function in reverse.
Logarithmic Functions: Scary or Easy?
Some of you may find the term logarithm or logarithmic function intimidating. Relax! In this lesson, we are going to demystify the term and show you how easy it is to work with logarithms. To keep things simple, we will use the term logs when referring to logarithms or a logarithmic function.
Exponents vs. Logarithms
An exponent is just a way to show repeated multiplication. For instance, in 32 = 3*3 = 9, the 3 is called the base of the exponent and the superscripted 2 is called the exponent or power. An exponential function tells us how many times to multiply the base by itself. Here are some examples:
53 = 5*5*5 = 25*5 =125 means take the base 5 and multiply it by itself three times.
For 25, we take the 2 and multiply it by itself five times, like this: 2*2*2*2*2 = 4*2*2*2 = 8*2*2 = 16*2 = 32. This yields a result of 32.
An exponential function is written this way, where b is the base and x is the exponent.
A logarithmic or log function is the inverse of an exponential function. We can use a log function to find an exponent. Let's use this information to set up our log. The logarithmic function is written this way:
Notice that the b is the same in both the exponential function and the log function and represents the base. Here is an example of using the same set of information and expressing it as a log and an exponent:
Exponential function form: 32 = 9
Logarithmic function form: log base 3 of 9 = 2
Stop and take a look at both forms. In exponential function form, we have 9 as the answer. In the log form, the 2 is the answer and represents the exponent. What did we say was a log? A log is an exponent or in another format: log = exponent. Let's try a few more.
Notice on the last logarithm that we did not include the base 10. Base 10 is called the common log. This is one of the most often used logs and is the base on all calculators with a log button. If you see a log written without a base, this is base 10.
To make sure you understand how to go from logs to exponents and back, try these:
Write as an exponent: log base 10 of 100 = 2
Write as an exponent: log base 5 of 25 = 2
Write as a log: 93 = 729
Write as a log: 62 = 36
Write as a log: 100= 1
Write as an exponent: log base 3 of 1/27 = -3
102 = 100
52 = 25
log base 9 of 729 = 3
log base 6 of 36 = 2
Log base 10 of 10 = 1
3-3 = 1/27
Did you stumble on the last one? Remember exponents can also be negatives. A negative exponent just means the reciprocal. So 3-3 = 1/33 = 1/27.
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The logarithmic function is used in many areas of study, from engineering to earthquake measurement to anything that has to do with growth and decay, like investments or radioactivity. For instance, in 1989, the third game of the Battle of the Bays World Series in San Francisco, CA was suddenly interrupted by a very strong earthquake. The Loma Prieta earthquake measured 7.1 on the Richter scale. It stopped the World Series as two bridges in San Francisco collapsed and buildings shook violently, causing an estimated $6 billion in property damages.
Logarithms are used to find the intensity of the earthquakes. The formula for finding the Richter scale of an earthquake is found by:
I0 is the intensity of an earthquake that is barely felt, a zero-level earthquake. I is the intensity of the earthquake and R is the Richter scale value.
To find the intensity of the Loma Prieta earthquake, let's plug in the value: 7.1 = log I (the I0 cancels). Remember the 7.1 is the exponent on the base 10. If we were to rewrite this log as an exponent, it would look like this: 107.1 = I. The intensity of this earthquake was 107.1 = 12,589,254.12. This means the earthquake was 12.5 million times as intense as a zero-level earthquake.
Natural Logarithms Used with Base E
Another logarithm, called the natural logarithm, is used when dealing with growth and decay. The base of a natural logarithm is the natural number, e, which is an irrational number. This means that e cannot be perfectly represented in base 10, since it is a decimal that does not terminate. Natural logarithms are written as ln and pronounced as log base e. The only difference between a natural logarithm and a common logarithm is the base. The natural log has base e, which is approximately 2.718. The common logarithm has base 10.
We can still translate the natural log as the exponent on base e. So if you had e2 = 7.389, you could write it as a natural log: ln 7.389 = 2. The base is e, and the exponent is the answer. Here are a few examples:
A logarithmic function is the inverse of an exponential function. The base in a log function and an exponential function are the same. A logarithm is an exponent. The exponential function is written as: f(x) = bx. The logarithmic function is written as: f(x) = log base b of x. The common log uses the base 10. The natural log uses the base e, which is an irrational number, e = 2.71828.
A logarithm is an exponent. Any exponential expression can be rewritten in logarithmic form. For example, if we have 8 = 23, then the base is 2, the exponent is 3, and the result is 8. This can be rewritten in logarithmic form as
3 = log2 8.
Notice how the numbers have been rearranged. The subscript on the logarithm is the base, the number on the left side of the equation is the exponent, and the number next to the logarithm is the result (also called the argument of the logarithm). Now, try rewriting some of the following in logarithmic form:
Rewrite each of the following in logarithmic form:
1. 25 = 52
2. 64 = 43
3. 1 = 50
4. 1/4 = 2-2
1. 2 = log5 25
2. 3 = log4 64
3. 0 = log5 1
4.-2 = log2 1/4
Now, we can also start with an expression in logarithmic form, and rewrite it in exponential form. For example, the expression 3 = log5 125 can be rewritten as 125 = 53.
Here, 5 is the base, 3 is the exponent, and 125 is the result. Now try the following:
Rewrite each of the following in exponential form:
1. 4 = log2 16
2. 2 = log3 9
3. 1/2 = log9 3
4. 3/2 = log4 8
1. 24 = 16
2. 32 = 9
3. 91/2 = 3
4. 43/2 = 8
Now try solving some equations. For example, y = log2 8 can be rewritten as 2y = 8.
Since 8 = 23 , we get y = 3.
As mentioned in the beginning of this lesson, y represents the exponent, and it also represents the logarithm. Therefore, a logarithm is an exponent.
Calculate each of the following logarithms:
1. y = log5 125
2. y = log3 1
3. y = log9 27
4. y = log4 1/16
We could solve each logarithmic equation by converting it in exponential form and then solve the exponential equation. We have:
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