Change of Base Formula

A logarithm can have any positive base except , but a calculator only knows two of them: base behind the key and base behind the key. To find or , you need a way to translate a logarithm from one base to another. That translation is the change of base formula, and it is also the step that finishes off almost every exponential equation you will solve.

The Change of Base Formula

For positive numbers , and , with and :

The new base can be anything, so choose the one your calculator has. With base or base :

To remember the arrangement: the argument goes on top, the old base goes on the bottom. The number that was “up” in stays up; the small subscript base drops to the denominator.

Why the Change of Base Formula Works

The proof is a single application of the power rule.

Let . In exponential form that is . Take of both sides:

The power rule brings down to the front:

Dividing by isolates , which was all along:

How to Change the Base of a Logarithm

1. Write the argument’s logarithm in the numerator: (or ).
2. Write the old base’s logarithm in the denominator: (or ).
3. Evaluate both, then divide. Keep several decimal places until the final step.

For :

The answer makes sense: and , so the exponent that gives must be a little above .

Example 1: Evaluating Logarithms With Different Bases

Two sanity checks are worth making every time. The sign: the log is negative when the argument and base are on opposite sides of , as in the last two rows. The size: the answer should sit between the two whole-number powers of the base that bracket the argument.

Changing to a Friendlier Base for Exact Answers

The formula is not only for calculators. If the base and the argument are both powers of the same number, changing to that number gives an exact fraction.

Two Useful Consequences

Swapping base and argument gives the reciprocal. Choosing as the new base:

So and .

Logs chain together. Writing both factors over a common base, the middle cancels:

This is the identity to reach for when a problem multiplies logs with different bases.

Worked Example

Evaluate to four decimal places, and confirm that and give the same answer.

Step 1 — apply the formula with natural logs:

Step 2 — evaluate the natural logs:

Step 3 — repeat with common logs:

Step 4 — check the size: and , so a value just under is right, since is just under .

So , whichever key you use.

Common Mistakes to Avoid

• Putting the base on top. . That gives , which is — the reciprocal.
• Simplifying the fraction as a single log. is not and not . It is a quotient of two numbers; evaluate each and divide.
• Mixing log and ln in one fraction. gives the wrong answer. Use the same logarithm on top and bottom.
• Rounding too early. , but . Carry at least six decimal places until the final division.

Practice Problems

Work each one before opening the answer.

Problem 1. Evaluate to four decimal places.

Problem 2. Evaluate to four decimal places.

Problem 3. Find the exact value of .

Problem 4. Find the exact value of .

Problem 5. Show that .

Problem 6. Solve to four decimal places.

Problem 6 is a preview of the next step. The solving exponential equations lesson turns that one move — take a log, then change base if needed — into a method for any equation with the variable in the exponent.