Change of Base Formula
The change of base formula may be used when you may need to change bases to evaluate a specific logarithm. If a, b, and x are positive real numbers such that a and b do not equal one, then log base a can be converted to a different base.
Properties of Logarithms
When two logarithms with a common base is multiplied by two numbers, it can be rewritten as the sum of the two logarithms. This same idea goes for division. When two logarithms with a common base are divided by two numbers, the logarithm can be rewritten as the difference of the two logarithms. The third property of logarithms involves the exponent being put at the front of the logarithm as expressed by the third equation above.
Example
Using the first property of logarithms it is possible to convert the natural log of 6 into the natural log of 2 plus the natural log of 3.
Example 2
All properties of logarithms work whether you are expanding the logarithm or condensing it.
x=3
Proof
Below is a proof of the division property for logarithms
b/c=b/c
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