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There is an inverse relationship between logarithms and exponents. If we have the expression , we can gather that x equals 2, because 3 squared equals 9 ().
As a logarithmic expression, we can write this equivalently as . This reads, "the log, base 3, of 9 is 2." The expression tells us that the base number, 3, must be raised to the power of 2 in order to equal 9.
In general, we can write the relationship between logarithms and exponents as follows:
Exponential equation | |
Logarithmic equation |
Notice that x and y switched as being isolated onto one side of the equals sign. This is characteristic of inverse relationships. Also, note that the base to the exponential is the base of the logarithm.
EXAMPLE
Rewrite the exponential equation as a logarithmic equationExponential equation | |
Logarithmic equation |
If you have a scientific calculator that can compute logarithms, there are likely two kinds of log buttons on your calculator: one that simply says "log" and another that says "ln." The first button, "log" is known as the common log, while the other, "ln," is referred to as the natural log.
They are both logarithms, but their difference is in their base. Common log operates under a base of 10. So if you ever see expressions such as log(42) or log(67), the base of the log is 10.
The abbreviation "ln" comes from the Latin logarithmus naturali. The base of this logarithm is the mathematical constant "e". The constant "e", or Euler's constant, is approximately equal to 2.718282. If you have the natural log button (ln) on your calculator, definitely use it for the most accurate calculations. Otherwise, use the approximation 2.718282.
We can use the relationship between exponential equations and logarithmic equations to evaluate expressions by thinking about how many times we must multiply a given number by itself to result in another given number.
EXAMPLE
Evaluate .Rewrite using exponents | |
4 cubed results in 64 | |
Write the solution to expression | |
Our solution |
EXAMPLE
Evaluate .Rewrite using exponents | |
3 raised to the 5th power is 243 | |
Write the solution to expression | |
Our solution |
Notice how the bases are the same in both exponential and logarithmic form.
Source: ADAPTED FROM "BEGINNING AND INTERMEDIATE ALGEBRA" BY TYLER WALLACE, AN OPEN SOURCE TEXTBOOK AVAILABLE AT www.wallace.ccfaculty.org/book/book.html. License: Creative Commons Attribution 3.0 Unported License