A logarithm answers the question “what power gives this number?”: logₐ x = y means aʸ = x. So log₂ 8 = 3 because 2³ = 8. Logarithms are the inverse of exponentials, which means you can always switch between log form and power form, and the graph of y = logₐ x is the reflection of y = aˣ in the line y = x.
Logarithm basics
- logₐ x = y means aʸ = x (a > 0, a ≠ 1, x > 0)
- logₐ 1 = 0 and logₐ a = 1
- logₐ aᵏ = k
- log x on a calculator means log₁₀ x
- y = logₐ x passes through (1, 0) and has asymptote x = 0
What is a logarithm?
logₐ x asks: a to what power gives x? The small number a is the base. log₂ 8 asks “2 to what power gives 8?”, and the answer is 3. A logarithm undoes an exponential in the same way that square-rooting undoes squaring.
Switching between log form and power form
aʸ = x and logₐ x = y carry exactly the same information. When switching, the base stays the base, while the power and the result swap places: 3⁴ = 81 becomes log₃ 81 = 4. Being able to flip between the two forms is the most useful skill in the whole topic.
Special values and the domain
Three results come straight from the definition: logₐ 1 = 0 because a⁰ = 1, logₐ a = 1 because a¹ = a, and logₐ aᵏ = k. You can only take the logarithm of a positive number, so x > 0, but the answer can be any real number, including negative values.
Base 10, the common logarithm
Base 10 is used so often that the base is usually left out: log x means log₁₀ x, and it is the log button on a calculator. For powers of 10 you do not need a calculator: log 1000 = 3 and log 0.1 = −1.
The graph of y = logₐ x
Because the logarithm is the inverse of the exponential, y = logₐ x is the reflection of y = aˣ in the line y = x. Every feature swaps: the exponential passes through (0, 1), so the log graph passes through (1, 0); the exponential has asymptote y = 0, so the log graph has asymptote x = 0, the y-axis. The log graph exists only for x > 0.
Worked examples
Example 1. Evaluate log₂ 8.
- Ask: 2 to what power is 8?
- 2¹ = 2, 2² = 4, 2³ = 8.
Answer: log₂ 8 = 3.
Marks: M1 for 2³ = 8, A1 for 3.
Example 2. Write 3⁴ = 81 in logarithmic form.
- The base is 3.
- 81 goes inside the log, and the power 4 is the result.
Answer: log₃ 81 = 4.
Marks: A1 for log₃ 81 = 4.
Example 3. Evaluate log₇ 1, log₆ 6 and log₂ 2⁵.
- 7⁰ = 1, so log₇ 1 = 0.
- 6¹ = 6, so log₆ 6 = 1.
- The base matches the base of the power, so log₂ 2⁵ = 5.
Answer: 0, 1 and 5.
Marks: B1 for each value.
Example 4. Evaluate log 10 000 and log 0.01.
- 10 000 = 10⁴, so log 10 000 = 4.
- 0.01 = 10⁻², so log 0.01 = −2.
Answer: 4 and −2.
Marks: B1 for each value.
Example 5. State where y = log₂ x crosses the x-axis, and its asymptote.
- y = log₂ x is the reflection of y = 2ˣ in y = x.
- 2ˣ crosses the y-axis at (0, 1), so log₂ x crosses the x-axis at (1, 0).
- 2ˣ has asymptote y = 0, so log₂ x has asymptote x = 0.
Answer: It crosses at (1, 0); the asymptote is x = 0.
Marks: B1 for (1, 0), B1 for x = 0.
Common mistakes
- Swapping the base and the result. 3⁴ = 81 is log₃ 81 = 4, not log₄ 81 = 3 or log₃ 4 = 81.
- Taking the log of zero or a negative number. logₐ x is only defined for x > 0.
- Thinking logs cannot be negative. log 0.01 = −2; the answer can be any real number.
- Using the wrong calculator button. log means base 10; ln means base e.
- Drawing the log graph for negative x. y = logₐ x only exists for x > 0.
How logarithms are first examined
Logarithms are introduced in Pure 2 (WMA12) of Edexcel International A-Level, and in first-year Pure of Edexcel A-Level (9MA0). Early questions ask you to:
- write a power statement in logarithmic form, or the reverse
- evaluate logarithms exactly, including negative and fractional answers
- find an unknown base from a given logarithm
- sketch y = logₐ x and state its intercept and asymptote
Find these with mark schemes in the Pure 2 logarithms exam questions. The exponentials and logarithms worksheet and the revision engine for exponentials and logarithms give more practice.
Next come the laws of logarithms, which let you combine and split logs. This builds on exponential graphs.
Frequently asked questions
What does log₂ 8 mean?
The power you raise 2 to in order to get 8. Since 2³ = 8, log₂ 8 = 3.
Why can't you take the log of a negative number?
Because a positive base raised to any real power is always positive, so no power can give a negative number.
What is the difference between log and ln?
log usually means base 10, and ln means base e, the natural logarithm.