When the unknown is in a power, as in 3ˣ = 20, take logarithms of both sides and use the power law to bring the exponent down: x log 3 = log 20, so x = log 20 / log 3. The same idea handles linear exponents and the variable on both sides. A quadratic in aˣ is solved by substituting y = aˣ, and logarithmic equations are solved by combining the logs and converting to power form.
Methods for exponential and log equations
- aˣ = b gives x = log b / log a
- a^(px + q) = b gives (px + q) log a = log b
- Quadratic in aˣ: let y = aˣ, solve, then reject y ≤ 0
- logₐ f(x) = k gives f(x) = aᵏ
Check every solution of a log equation: each log must have a positive argument.
Taking logs of both sides
To solve aˣ = b, take logarithms of both sides. The power law brings the x down: x log a = log b, so x = log b / log a. Give the exact answer as a ratio of logs if asked, and a decimal to 3 significant figures otherwise. Either log or ln can be used, as long as you are consistent.
A linear expression in the power
If the power is an expression such as 2x + 1, the method is the same: bring the whole expression down in brackets, so (2x + 1) log 3 = log 40, then solve the linear equation. Keep the exact log value until the final rounding.
Hidden quadratics in aˣ
An equation such as 2²ˣ − 5 × 2ˣ + 4 = 0 is a quadratic in disguise, because 2²ˣ = (2ˣ)². Substitute y = 2ˣ to get y² − 5y + 4 = 0, solve it, then solve 2ˣ = y for each root. Since aˣ is always positive, reject any value of y that is zero or negative.
Isolating the power, and the variable on both sides
If the power is not on its own, rearrange first: 5 × 2ˣ = 160 becomes 2ˣ = 32, which may have an exact answer. If the variable appears in powers on both sides, take logs of both sides, expand the brackets, collect the x terms on one side and factorise x out.
Solving logarithmic equations
When the unknown is inside the logarithms, combine every log term into a single logarithm using the laws, writing any constant as a log if needed. Then convert to power form: logₐ f(x) = k means f(x) = aᵏ. Because a logarithm needs a positive argument, check each solution in the original equation and reject any that make an argument zero or negative.
Worked examples
Example 1. Solve 3ˣ = 20, giving your answer to 3 significant figures.
- Take logs: x log 3 = log 20.
- x = log 20 / log 3 = 2.7268…
Answer: x = 2.73 (3 s.f.).
Marks: M1 for x log 3 = log 20, A1 for 2.73.
Example 2. Solve 3²ˣ = 50 to 3 significant figures.
- Take logs: 2x log 3 = log 50.
- 2x = log 50 / log 3 = 3.5609…
- x = 1.7805…
Answer: x = 1.78 (3 s.f.).
Marks: M1 for 2x log 3 = log 50, A1 for 1.78.
Example 3. Solve 2²ˣ − 5 × 2ˣ + 4 = 0.
- Let y = 2ˣ: y² − 5y + 4 = 0.
- (y − 1)(y − 4) = 0, so y = 1 or y = 4.
- 2ˣ = 1 gives x = 0; 2ˣ = 4 gives x = 2.
Answer: x = 0 or x = 2.
Marks: M1 for the quadratic in y, A1 for y = 1, 4, A1 for both values of x.
Example 4. Solve 5 × 2ˣ = 160.
- Divide by 5: 2ˣ = 32.
- 32 = 2⁵.
Answer: x = 5.
Marks: M1 for 2ˣ = 32, A1 for x = 5.
Example 5. Solve log₂(3x + 2) − log₂(x − 1) = 3.
- Quotient law: log₂((3x + 2)/(x − 1)) = 3.
- Power form: (3x + 2)/(x − 1) = 2³ = 8.
- 3x + 2 = 8x − 8, so x = 2.
- Check: x − 1 = 1 > 0 and 3x + 2 = 8 > 0, so x = 2 is valid.
Answer: x = 2.
Marks: M1 for the quotient law, M1 for the power form, A1 for x = 2.
Common mistakes
- Taking logs of each term separately. log(a + b) is not log a + log b; isolate the power first.
- Not bracketing the exponent. log 3^(2x + 1) = (2x + 1) log 3, with the whole power in brackets.
- Keeping an impossible root. 2ˣ = −3 has no solution, because 2ˣ is always positive.
- Rounding too early. Keep exact logs until the final answer.
- Not checking log equation solutions. Reject any value that makes an argument of a log zero or negative.
How equations with logarithms are examined
Exponential and log equations appear on almost every WMA12 (Pure 2) paper in Edexcel International A-Level, and 9MA0 students learn them in Year 1. Typical questions:
- solve aˣ = b, or an equation with a linear exponent, to 3 significant figures
- solve a hidden quadratic in aˣ
- solve an equation where the logs must first be combined
- solve simultaneous equations involving logarithms
Mark-schemed questions on these equations are in the Pure 2 exponentials and logarithms exam question set. For extra practice use the exponentials and logarithms worksheet or the A-Level exponentials and logarithms revision engine.
You need the laws of logarithms throughout. Equations with different bases are covered in changing the base.
Frequently asked questions
How do you solve an equation like 3ˣ = 20?
Take logs of both sides: x log 3 = log 20, so x = log 20 / log 3 ≈ 2.73.
Should I use log or ln?
Either works for solving aˣ = b, as long as you use the same one on both sides.
Why do some solutions of log equations get rejected?
A logarithm is only defined for positive numbers, so any solution that makes an argument zero or negative is invalid.