The Change of Base Formula

The change of base formula is given as:

$$\begin{align}& log_{b}(x) = \frac{log_{d}(x)}{log_{d}(b)} \end{align}$$

Where b is the original base of the logarithm, and d is the base that we are changing to.

Why do we Perform a Change of Base?

Sometimes we are faced with a logarithm that we are unable to evaluate given its current base. For example, we may have to calculate log₆(3) but our calculator can only perform base 10 logarithms. This is where the change of base formula can come in and save the day.

Another situation where it proves to be extremely useful is when trying to solve logarithms by hand. There are certain combinations of logarithm base and logarithm subject that can be solved without the use of a calculator. The change of base formula can help us format a logarithm into one of these combinations.

Rules to Follow

Using the change of base formula allows us to calculate a logarithm of any base b, with restrictions that b > 0 and b ≠ 1. These restrictions are in place because if b ≤ 0 or b = 1, the result will be indeterminate (meaning we will be unable to get the answer).

Change of Base Example Problem

Let's work through an example problem together to practice using the change of base formula.

Evaluate log₄(12) as if the calculator is only able to compute logarithms of base e (Euler's number).

Solution:

1. Let’s set b = 4, x = 12, and d = e. Using e as our logarithm base for evaluation allows us to use the natural logarithm, notated as ln.
2. Plugging our constants into the change of base formula gives us:

$$\begin{align}& log_{4}(12) = \frac{log_{e}(12)}{log_{e}(4)} = \frac{ln(12)}{ln(4)} = \frac{2.4849}{1.3863} = 1.7925 \end{align}$$

3. The logarithm log₄(12) evaluated as ^ln(12)/_ln(4) results in 1.7925.