What is a logarithm? A logarithmic equation is another way to write an exponential equation. Recall from the previous section the form of an exponential equation:
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when
rewritten becomes |
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In other words, in the exponential equation, x is the input, and it is located in the exponential position, whereas the logarithmic equation is set up so that the output is x , the output in this case being the power needed to raise the base a to, to get a number y.
So what is a logarithm? A logarithm is the exponent needed to raise a base number to, to get the desired output number. Huh? By going through the examples below, you will make sense of this.
Example 1 - How do you convert exponential equations to logarithmic equations?
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Exponential Equation |
Equivalent Logarithmic Equation |
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10 3 = 1000 |
log 101000 = 3 or just log 1000 = 3 (base 10 is called the common log) |
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10 -2 = 0.01 |
log 0.01 = -2 |
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5 2 = 25 |
log 5 25 = 2 |
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y = 2 x |
log 2 y = x |
So you see, in each case, the logarithm is the power that we raised the base to, to get the numbers 1000, .01, etc.
Example 2 - How do you convert logarithmic equations to exponential equations?
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Logarithmic Equation |
Equivalent Exponential Equation |
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ln e = 1 or log e e = 1 |
e1 = 2.71828... |
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log1 = 0 or log 10 1 = 0 |
10 0 = 1 |
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log 2 8 = 3 |
2 3 = 8 |
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log 7 x = y |
7 y = x |
When the process is reversed, the logarithm becomes the exponent or power once again, needed to raise the base to, to get the desired numerical result, i.e., 2.718, 1, etc. Logarithms may have given you trouble in college algebra. There is just no substitute for working a lot of small simple problems to alleviate this. A handout prepared by a colleague called “Become Friends with Logarithms” provides the kind of simple, repetitive practice that will clarify this concept of a logarithm. Click here to get a copy of the handout.
Example 3 - What does a logarithmic graph look like?
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We have observed that the typical exponential function increases quite rapidly. If we switch the input and the output in the function, we would expect the reverse to happen. We would expect the "reversed" graph to increase very slowly. This "reversed" graph (called the inverse graph) does indeed increase quite slowly. This is the typical logarithmic function. |
Example 4 - Properties of logarithms.
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The properties of logarithms are the same as the properties of exponents, since a logarithm is an exponent. However, written in logarithmic notation, they look quite different. You should have a working familiarity with these properties. The properties in the box are linked to another site where you can practice problems involving logarithms. |
Use the above properties of logarithms to solve the following problems:
a) You deposit a sum of money in the bank. Assuming continuous compounding with an interest rate of 7%, how long will it take the money to double in size?
solution
b) You deposit a sum of money in the bank. Assuming annual compounding with an interest rate of 7% , how long will it take the money to double in size?
solution
Mat210 Section 1.11b Derivatives of Exponential and Logarithmic Functions
Last Update: January 9, 2010 Leslie Arce (c) 2010 Sharon Walker and the ASU Department of Mathematics - All rights reserved.
which says in English, "The
logarithm of y to the base a is x."
