Logarithms (Advanced)

A logarithm is an exponent: logₐ x = y means aʸ = x. The laws of logarithms turn products into sums and help solve equations.

Key ideas

logₐ x is defined only for x > 0, with base a > 0 and a ≠ 1.
To evaluate a log exactly, write both the base and the number as powers of the same number: log₈ 4 = 2/3, since 8 = 2³ and 4 = 2².
Combine logs into one using the laws, then switch to exponential form to solve an equation.
Always check every root: a value that makes any argument zero or negative is extraneous and is rejected.
A quadratic in logₐ x is solved by putting y = logₐ x first.
Common logs (base 10) measure scales: one Richter unit, or one pH step, is a factor of 10.

Definition	logₐ x = y ⇔ aʸ = x
Product and quotient	logₐ (MN) = logₐ M + logₐ N; logₐ (M/N) = logₐ M − logₐ N
Power	logₐ (Mᵏ) = k logₐ M
Special values	logₐ 1 = 0; logₐ a = 1
Change of base (any base k)	logₐ x = logₖ x ÷ logₖ a; so logₘ n × log_n p = logₘ p
pH	pH = −log₁₀ [H⁺]

Writing logₐ (M + N) = logₐ M + logₐ N: the product law is for M × N, not M + N.
Keeping an extraneous root such as x = −1, which would need the log of a negative number.
Writing (logₐ M)ᵏ for logₐ (Mᵏ): only the power inside the log comes down in front.

Solve log₁₀ (x + 8) + log₁₀ (x − 187) = 3, and check each root.

Working: (x + 8)(x − 187) = 1000; x² − 179x − 2496 = 0; (x − 192)(x + 13) = 0; x = −13 makes both arguments negative, so it is rejected

Answer: x = 192