Solving for a variable inside a log
A logarithmic equation has the variable in the argument (inside) of a logarithm:
- log₂ x = 5
- log(x + 3) = 2
- log₃ x + log₃(x − 8) = 2
Method 1: Convert to exponential form
If one log equals a number, rewrite in exponential form.
Example: log₂ x = 5 → x = 2⁵ = 32
Example: log(2x − 4) = 2 → 2x − 4 = 10² = 100 → x = 52
Method 2: Condense, then convert
Use the log laws to condense to a single log first.
Example: log₃ x + log₃(x − 8) = 2
- log₃[x(x − 8)] = 2
- x² − 8x = 9
- x² − 8x − 9 = 0 → (x − 9)(x + 1) = 0
- x = 9 or x = −1
Check for extraneous solutions
The argument of a log must be positive.
- x = 9: log₃ 9 + log₃ 1 = 2 + 0 = 2 ✓
- x = −1: log₃(−1) is undefined ✗
x = −1 is an extraneous solution. Answer: x = 9.
Method 3: One-to-one property
If both sides are single logs with the same base, set the arguments equal (one-to-one property).
Example: log(x + 7) = log(3x − 1)
- x + 7 = 3x − 1 → x = 4
- Check: log 11 = log 11 ✓
Example: 2 log x = log 36
- log x² = log 36 → x² = 36 → x = ±6
- Reject −6 (log of a negative) → x = 6
Applications
pH: pH = −log[H⁺]. If pH = 3.5, then [H⁺] = 10^(−3.5) ≈ 3.16 × 10⁻⁴ mol/L.
Sound: L = 10 log(I/I₀). A 70 dB sound has I/I₀ = 10⁷.
Earthquakes: each whole-number increase in magnitude means about 10 times the ground motion.
Worked examples
Example 1: log₅(x + 1) = 2 → x + 1 = 25 → x = 24.
Example 2: log x − log 2 = 1 → log(x/2) = 1 → x/2 = 10 → x = 20.
Example 3: log₂ x + log₂(x + 2) = 3 → x² + 2x − 8 = 0 → x = 2 or −4 → x = 2.
Common mistakes
- Forgetting to check that arguments are positive.
- Using log laws incorrectly, like log x + log y = log(x + y).
- Converting to exponential form with the base and exponent swapped.
Try it yourself
- log₄ x = 3
- log(x − 5) = 1
- log₂ x + log₂ 4 = 5
- log₆ x + log₆(x − 5) = 2
- log(2x + 1) = log(x + 4)
Answers: 1) 64 2) 15 3) 8 4) 9 (−4 is extraneous) 5) 3
Quick recap
- Convert single logs to exponential form.
- Condense sums and differences first.
- Reject extraneous solutions: arguments must be positive.