The change of base formula rewrites a logarithm in any base you choose: logₐ x = logₖ x / logₖ a. Using base 10 lets a calculator evaluate any logarithm, and choosing a shared base gives exact answers. A useful consequence is that logₐ x = 1 / logₓ a, which turns equations mixing logₐ x and logₓ a into quadratics.
Change of base
- logₐ x = logₖ x / logₖ a, for any base k
- logₐ x = log x / log a (using base 10)
- logₐ x = 1 / logₓ a
- logₐ x × logₓ a = 1
The change of base formula
Any logarithm can be rewritten in a different base k: logₐ x = logₖ x / logₖ a. Choosing base 10 means the calculator's log button can evaluate logarithms in any base. The formula is in the formula booklet, but it is worth understanding: it comes from writing x = aʸ and taking logs of both sides.
Exact values using a shared base
When the base and the number are both powers of the same value, change to that value as the new base and the answer comes out exactly. For example, 4 = 2² and 8 = 2³, so log₄ 8 = log₂ 8 / log₂ 4 = 3/2. Spotting the shared base first saves a calculator approximation.
The reciprocal relationship
Changing to the argument x as the new base gives logₐ x = logₓ x / logₓ a = 1 / logₓ a. So a logarithm and its “flip” are reciprocals, and their product is 1. This is the key to the hardest questions, where logₐ x and logₓ a appear in the same equation.
Equations with different bases
If an equation contains logarithms to different bases, change them all to one common base, usually the smallest, and then solve as usual. For example, log₄ 9 = log₂ 9 / log₂ 4 = (2 log₂ 3)/2 = log₂ 3.
Hidden quadratics using the reciprocal rule
An equation such as log₂ x + 2 logₓ 2 = 3 mixes a log with its flip. Let y = log₂ x, so logₓ 2 = 1/y. Multiply through by y to get a quadratic, solve it, then convert back with x = 2ʸ. Reject any value of x that is not positive or that equals 1, because a base must be positive and not 1.
Worked examples
Example 1. Evaluate log₂ 10 to 3 significant figures.
- Change to base 10: log₂ 10 = log 10 / log 2.
- log 10 = 1, so this is 1 / log 2 = 3.3219…
Answer: 3.32 (3 s.f.).
Marks: M1 for log 10 / log 2, A1 for 3.32.
Example 2. Evaluate log₄ 8 exactly.
- Change to base 2: log₄ 8 = log₂ 8 / log₂ 4.
- log₂ 8 = 3 and log₂ 4 = 2.
Answer: 3/2.
Marks: M1 for the change of base, A1 for 3/2.
Example 3. Given log₂ 5 ≈ 2.32, find log₅ 2 to 3 significant figures.
- Use the reciprocal rule: log₅ 2 = 1 / log₂ 5.
- 1 / 2.32 = 0.4310…
Answer: 0.431 (3 s.f.).
Marks: M1 for 1 / log₂ 5, A1 for 0.431.
Example 4. Solve log₂ x = log₄ 9.
- Change the right-hand side to base 2: log₄ 9 = log₂ 9 / 2.
- log₂ 9 / 2 = 2 log₂ 3 / 2 = log₂ 3.
- So log₂ x = log₂ 3.
Answer: x = 3.
Marks: M1 for the change of base, A1 for x = 3.
Example 5. Solve log₂ x + 2 logₓ 2 = 3.
- Let y = log₂ x, so logₓ 2 = 1/y: y + 2/y = 3.
- Multiply by y: y² − 3y + 2 = 0, so (y − 1)(y − 2) = 0.
- y = 1 gives x = 2; y = 2 gives x = 4.
Answer: x = 2 or x = 4.
Marks: M1 for y + 2/y = 3, M1 for the quadratic, A1 for both values.
Common mistakes
- Turning the fraction upside down. logₐ x = log x / log a, with the base on the bottom.
- Mixing bases in one equation. Change every log to a single common base before solving.
- Rounding too early. Keep exact forms, such as log₂ 3, as long as you can.
- Forgetting to check the base condition. In logₓ a, x must be positive and not equal to 1.
- Missing a shared base. Look for both numbers being powers of the same base for an exact answer.
How change of base is examined
Change of base completes the logarithms chapter of Pure 2 (WMA12) in Edexcel International A-Level; in Edexcel A-Level (9MA0) it is covered in Year 1. Questions ask you to:
- evaluate a logarithm in an unusual base to 3 significant figures
- find an exact value by changing to a shared base
- simplify a product of logarithms with different bases
- solve equations that mix bases, including hidden quadratics
Change of base questions, with full mark schemes, appear in the Pure 2 log and exponential exam questions. The exponentials and logarithms worksheet and the exponentials and logarithms revision engine give more practice.
This builds on the laws of logarithms and solving equations with logs.
Frequently asked questions
What is the change of base formula?
logₐ x = logₖ x / logₖ a for any positive base k ≠ 1. With base 10, logₐ x = log x / log a.
How do you evaluate log₇ 20 on a calculator?
Use log 20 / log 7 ≈ 1.54, or ln 20 / ln 7, which gives the same answer.
Why is logₐ x equal to 1 / logₓ a?
Use the change of base formula with x as the new base: logₐ x = logₓ x / logₓ a = 1 / logₓ a, because logₓ x = 1.