Exponential graphs and logarithms

The graph of y = a^x (a > 0, a not 1)

• Passes through (0, 1)
• Always positive; the x-axis (y = 0) is an asymptote
• a > 1: increasing; 0 < a < 1: decreasing

What a logarithm is

log_a b = x means a^x = b. So log_2 8 = 3 because 2³ = 8.

Useful facts: log_a a = 1 and log_a 1 = 0. You cannot take the log of 0 or a negative number.

Inverse functions

y = log_a x is the inverse of y = a^x. Its graph is the reflection of y = a^x in the line y = x, passing through (1, 0).

Worked example

Find log_2 32.

1. Ask: 2 to what power gives 32?
2. 2⁵ = 32

Answer: log_2 32 = 5

Check yourself

1. Find log_3 81.
   Show answer

   4. 3⁴ = 81, so log_3 81 = 4.

2. Find log_5 (1/25).
   Show answer

   -2. 5^-2 = 1/25, so the answer is -2.

3. Find log_4 2.
   Show answer

   0.5. 4^1/2 = 2, so log_4 2 = 0.5.

4. Solve log_2 x = 3.
   Show answer

   8. x = 2³ = 8.

5. Write 10³ = 1000 in logarithm form.
   • log_3 1000 = 10
   • log_1000 10 = 3
   • log_10 1000 = 3
   • log_10 3 = 1000
   Show answer

   log_10 1000 = 3. a^x = b becomes log_a b = x, with a = 10, x = 3 and b = 1000.

6. Which is true for the graph of y = (1/2)^x?
   • It is decreasing and passes through (0, 1)
   • It is increasing and passes through (0, 1)
   • It is decreasing and passes through (1, 0)
   • It crosses the x-axis at x = 0
   Show answer

   It is decreasing and passes through (0, 1). 0 < a < 1 gives a decreasing curve, and a⁰ = 1.

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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.