An exponential statement such as 2⁵ = 32 and a logarithmic statement such as log₂ 32 = 5 say exactly the same thing. The logarithm is the index, the base stays the base, and the answer of the power becomes the number inside the log.
This conversion is the first skill in exponential and logarithmic reasoning. It is used whenever an unknown sits in an index, such as 3ˣ = 20, or whenever a log equation needs to be turned into an ordinary equation.
How do the two forms match up?
Take the pattern aˣ = y, which says “base a, raised to the power x, gives y”.
The logarithmic form is x = loga y. In words: “x is the power you put on a to get y”.
| Exponential form | Logarithmic form |
|---|---|
| 2⁵ = 32 | log₂ 32 = 5 |
| 10³ = 1000 | lg 1000 = 3 |
| 91/2 = 3 | log₉ 3 = 1/2 |
| 5⁻² = 1/25 | log₅ (1/25) = −2 |
| e³ = y | ln y = 3 |
Three things never move: the base stays the base, the index becomes the value of the log, and the number y becomes the number inside the log. Because a positive base raised to any power gives a positive result, you can only take the log of a positive number.
Worked example
(a) Find log₄ 64.
Ask: which power of 4 gives 64? Since 4³ = 64, log₄ 64 = 3.
(b) Solve logx 81 = 4.
Step 1, write as an index: x⁴ = 81.
Step 2, solve: x = 3, because 3⁴ = 81. The base must be positive, so the negative fourth root is not allowed.
Check: log₃ 81 asks “which power of 3 gives 81?” and the answer is 4. ✓
(c) Solve 5ˣ = 40, giving x to 3 significant figures.
Step 1, write as a log: x = log₅ 40.
Step 2, evaluate on a calculator using ln 40 ÷ ln 5 (a change of base, which your calculator may also do directly). This gives x = 2.292…
Answer: x = 2.29.
Check: 5² = 25 and 5³ = 125, so the power that gives 40 must lie between 2 and 3, and nearer 2. 2.29 fits.
The mistake to watch for
The usual slip is to swap the base with the answer of the power.
Mistaken conversion: from log₂ 8 = 3, a student writes 3² = 8.
The base 2 and the log 3 have traded places. 3² = 9, so the statement is false.
The correction is to say it aloud: “base 2, to the power 3, gives 8”, so 2³ = 8. The base is always the small number written beside “log”. A quick test is to substitute back: if the new statement is false with actual numbers, the conversion is wrong.
Check yourself
Try these, then open each answer. Use the non-calculator working trainer afterwards if you want to rehearse exact values.
1. Write 10⁴ = 10 000 as a logarithmic statement.
Show answer
The base is 10, the index is 4 and the number is 10 000. So lg 10 000 = 4, which means log₁₀ 10 000 = 4.
2. Find log₅ (1/25).
Show answer
Which power of 5 gives 1/25? Since 5² = 25, 5⁻² = 1/25. So log₅ (1/25) = −2.
3. Solve logx 8 = 3/2.
Show answer
Write x3/2 = 8. Raise both sides to the power 2/3: x = 82/3 = (∛8)² = 2² = 4.
Check: 43/2 = (√4)³ = 2³ = 8. So x = 4.
Where this leads next
With conversion secure, the next step is to combine logarithms with correct coefficients, because most log equations need a single log before you can convert. Then try the mixed practice set.
Some students follow every line in class yet lose marks when the unknown sits in the base or the index. A teacher working through your own solutions in online one-to-one Additional Mathematics tuition can find that pattern quickly.