Changing the Base of a Logarithm
The change of base formula allows you to rewrite a logarithm using a different base. The formula is
$$ \log_c x = \frac{ \log_b x }{ \log_b c } $$
where b is the original base and c is the new base.
This formula is especially useful when your calculator or software only works with specific logarithmic bases, such as base 10 or the natural logarithm.
In many textbooks, the same formula is also written in the following equivalent form
$$ \log_b x = \frac{ 1 }{ \log_c b } \cdot \log_c x $$
The quantity 1/logc(b) is called the conversion factor.
A practical example
Suppose you want to rewrite the logarithm of 15 in base 3 as a logarithm in base 5.
$$ \log_3 15 $$
To do this, apply the change of base formula
$$ \log_c x = \frac{ \log_b x }{ \log_b c } $$
In this example:
x = 15
b = 3
c = 5
Substituting these values into the formula gives
$$ \log_5 15 = \frac{ \log_3 15 }{ \log_3 5 } $$
You have now rewritten the logarithm of 15 using base 5 instead of base 3.
Proof of the formula
The proof starts from the definition of a logarithm.
A logarithmic equation can always be rewritten in exponential form:
$$ x = \log_c a \Leftrightarrow a=c^x $$
So the logarithm x=logc a is equivalent to the exponential equation
$$ a=c^x $$
Now take the logarithm in base b of both sides:
$$ \log_b a = \log_b c^x $$
Using the property of the logarithm of a power, the exponent x can be moved in front of the logarithm:
$$ \log_b a = x \cdot \log_b c $$
Next, solve the equation for x:
$$ x = \frac{\log_b a}{\log_b c} $$
Since x=logc a, substitute back into the equation:
$$ \log_c a = \frac{\log_b a}{\log_b c} $$
This is the change of base formula for logarithms.
