Logarithms • Topic 3 of 4

Logarithmic Equations

A logarithmic equation hides an unknown inside a log; the goal is to strip the log away. The standard playbook: combine all logs on one side into a single log using the laws, then convert log_b X = k into the exponential form X = b^k, or set two equal logs equal in their arguments. Substitution is the second big move — let t = log_b x to turn a log-quadratic like (log x)^2 − 3·log x + 2 = 0 into an ordinary quadratic in t, solve, then back-substitute. The non-negotiable final step in CAT is checking the domain: a logarithm is only defined for a positive argument and a positive base ≠ 1. Extraneous roots that make any log of a non-positive number must be rejected. Many CAT log equations are deliberately built so one of the two algebraic roots is invalid — testing whether you verify.

✅ Solved examples

1. Solve log_2(x − 3) = 4.
Exponential form: x − 3 = 2^4 = 16 ⇒ x = 19. Check: x − 3 = 16 > 0, valid.
2. Solve log x + log(x − 3) = 1 (base 10).
log[x(x−3)] = 1 ⇒ x(x−3) = 10 ⇒ x² − 3x − 10 = 0 ⇒ (x−5)(x+2)=0. x = 5 or −2; reject −2 (log of negative). x = 5.
3. Solve (log_2 x)² − 5·log_2 x + 6 = 0.
Let t = log_2 x: t² − 5t + 6 = 0 ⇒ (t−2)(t−3)=0 ⇒ t = 2 or 3 ⇒ x = 2^2 = 4 or 2^3 = 8.
4. Solve log_x 49 = 2.
x^2 = 49 ⇒ x = 7 (base must be positive, so reject −7). x = 7.

✏️ Practice — try these, take hints as needed

1. Solve log_3(2x + 1) = 2.
Convert to exponential form.
2x + 1 = 3^2 = 9.
Solve for x.
x = 4
2. Solve log_5 x + log_5(x − 4) = 1.
Combine: log_5[x(x−4)] = 1.
x(x−4) = 5.
Solve x² − 4x − 5 = 0, reject negative.
x = 5
3. Solve (log x)² − 4·log x + 3 = 0 (base 10).
Let t = log x.
t² − 4t + 3 = 0 ⇒ t = 1 or 3.
x = 10^t.
x = 10 or 1000
4. Solve log_2 x + log_4 x = 3.
log_4 x = (log_2 x)/2.
log_2 x (1 + 1/2) = 3.
(3/2)·log_2 x = 3 ⇒ log_2 x = 2.
x = 4
5. Solve log_x 81 = 4.
x^4 = 81.
81 = 3^4.
Base must be positive.
x = 3

📝 Topic test — 8 questions

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