Solving exponential equations

Taking logs

a^x = b gives x log a = log b, so x = log b / log a. Any base works, as long as you use the same one on both sides; your calculator's log button is base 10.

Change of base

log_a b = log_c b / log_c a. This lets you evaluate logs in any base on a calculator: log_3 50 = log 50 / log 3.

Hidden quadratics

2^2x = (2^x)². So 2^2x - 6(2^x) + 8 = 0 becomes y² - 6y + 8 = 0 with y = 2^x. Solve for y, then for x. Remember 2^x > 0, so reject negative values of y.

Worked example

Solve 3^x = 20, giving x to 3 decimal places.

1. Take logs: x log 3 = log 20
2. x = log 20 / log 3

Answer: x = 2.727

Check yourself

1. Solve 5^x = 40, giving x to 3 decimal places.
   Show answer

   2.292. x = log 40 / log 5 = 2.292.

2. Find log_3 50 to 3 decimal places.
   Show answer

   3.561. log_3 50 = log 50 / log 3 = 3.561.

3. Solve 2^{x + 1} = 7, giving x to 3 decimal places.
   Show answer

   1.807. (x + 1) log 2 = log 7, so x = log 7 / log 2 - 1 = 1.807.

4. Solve 4^x = 3^{x + 1}, giving x to 3 decimal places.
   Show answer

   3.819. x log 4 = (x + 1) log 3, so x(log 4 - log 3) = log 3 and x = 3.819.

5. Solve 2^2x - 6(2^x) + 8 = 0.
   Show answer

   x = 1 or x = 2. With y = 2^x: y² - 6y + 8 = 0, so y = 2 or 4. Then 2^x = 2 gives x = 1, and 2^x = 4 gives x = 2.

6. Solving 3^2x - 4(3^x) - 5 = 0 gives (3^x - 5)(3^x + 1) = 0. How many real solutions are there?
   • One, because 3^x = -1 has no solution
   • Two, one from each bracket
   • None
   • Two: x = 5 and x = -1
   Show answer

   One, because 3^x = -1 has no solution. 3^x = 5 gives x = log 5 / log 3. 3^x is always positive, so 3^x = -1 is impossible.

7. An investment is worth 500(1.04)^t pounds after t years. Find the least whole number of years for it to be worth more than 800 pounds.
   Show answer

   12. 1.04^t > 1.6, so t > log 1.6 / log 1.04 = 11.98. The least whole number is 12.

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Sources

• Pearson Edexcel International Advanced Level in Mathematics specification (issue 3)
• Pearson Edexcel IAL Pure Mathematics 2 scheme of work

Last updated 7 November 2026. Spotted a mistake? Tell us.