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Logarithm Practice Problems

Properties, equations, and change of base — with solutions.

A logarithm answers one question: what exponent turns the base into this number? Every log rule is an exponent rule read backwards, which is why students who memorize the rules without that translation invent fake ones — splitting log(a + b), or turning a log of a quotient into a quotient of logs. The fastest fix is deliberate practice on the real rules:

Product, quotient, power. Logs turn multiplication into addition, division into subtraction, and exponents into multipliers out front.
Domain. You can only take the log of a positive number — every log equation ends with a domain check, and every log function's domain comes from setting its argument greater than zero.
Solving equations. One log? Rewrite in exponential form. A log on each side with the same base? Set the arguments equal. An exponential equation? Take a log of both sides — or spot the hidden quadratic when the variable appears in two exponents.
Change of base. Any log is a ratio of logs in a base your calculator has — and, later, the reason calculus only ever needs ln.

In AP Calculus, logs never leave: ln x has the cleanest derivative in the course, and exponential growth problems are solved by taking logarithms. The eight problems below cover the full checklist, each with a worked solution.

Practice problems

Problem 1

Find the domain of

f (x) = lo g_{3} (3 x + 2)

Problem 2

Find the horizontal asymptote of

f (x) = 6 + 2 \cdot 2^{- x}

Problem 3

Solve

lo g_{3} (3 x - 6) = 0

for

x

Problem 4

Solve

lo g_{4} (4 x + 15) = lo g_{4} (8 - 3 x)

for

x

Problem 5

Which expression is equivalent to

lo g_{10} (\frac{M ^{3}}{N Q})

for all positive

M, N, Q

Problem 6

2^{3 x - 2} = 53

Solve for

x

Problem 7

2^{2 x} - 2^{x} - 2 = 0

Solve for

x

Problem 8

Which of the following is NOT equal to

lo g_{11} (17)

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