Change of base formula
The change of base formula is a formula for expressing a logarithm in one base in terms of logarithms in other bases.
For any positive real numbers such that neither nor are , we have
This allows us to rewrite a logarithm in base in terms of logarithms in any base . This formula can also be written
Use for computations
The change of base formula is useful for simplifying certain computations involving logarithms. For example, we have by the change of base formula that
Special cases and consequences
Many other logarithm rules can be written in terms of the change of base formula. For example, we have that . Using the second form of the change of base formula gives .
One consequence of the change of base formula is that for positive constants , the functions and differ by a constant factor, for all .
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