Logarithms — Hard Practice Quiz

A Algebra cheat sheet for Logarithms — every key formula with its symbols defined — plus a hard-level practice quiz to test recall.

Formulas & key concepts

Product Rule

$$\log_a(xy) = \log_a x + \log_a y$$

Quotient Rule

$$\log_a\left(\frac{x}{y}\right) = \log_a x - \log_a y$$

Power Rule

$$\log_a(x^n) = n \log_a x$$

Change of Base Formula

$$\log_a x = \frac{\log_c x}{\log_c a}$$

Inverse Property

$$a^{\log_a x} = x$$

Practice quiz

  1. Simplify the expression $2 \log_b x + \frac{1}{3} \log_b y - \log_b z^2$.

    • $\log_b \left( \frac{x^2 \sqrt[3]{y}}{z^2} \right)$
    • $\log_b \left( \frac{x^2 y^3}{z^2} \right)$
    • $\log_b \left( \frac{x^2 y^{1/3}}{z} \right)$
    • $\log_b (x^2 y^{1/3} z^2)$

    Answer: $\log_b \left( \frac{x^2 \sqrt[3]{y}}{z^2} \right)$

  2. Solve for $x$: $\log_3(x+6) + \log_3(x-2) = 2$.

    • $x=3$
    • $x=-7$
    • $x=3$ and $x=-7$
    • No real solution

    Answer: $x=3$

  3. Given $\log_a 2 = P$ and $\log_a 3 = Q$, express $\log_6 72$ in terms of $P$ and $Q$.

    • $\frac{3P+2Q}{P+Q}$
    • $\frac{2P+3Q}{P+Q}$
    • $3P+2Q - (P+Q)$
    • $\frac{P^3 Q^2}{PQ}$

    Answer: $\frac{3P+2Q}{P+Q}$

  4. If $\log_x y = A$ and $\log_y z = B$, express $\log_z x$ in terms of $A$ and $B$.

    • $\frac{1}{AB}$
    • $AB$
    • $A+B$
    • $\frac{A}{B}$

    Answer: $\frac{1}{AB}$

  5. Simplify $3^{2 \log_3 5 + \log_3 2}$.

    • $50$
    • $250$
    • $30$
    • $10$

    Answer: $50$

  6. Solve for $x$: $5^{2x-1} = 12$. Round your answer to 3 decimal places.

    • $1.272$
    • $0.272$
    • $1.700$
    • $1.544$

    Answer: $1.272$

  7. Simplify $\log_a \left( \frac{a^5 \cdot \sqrt[3]{b}}{c^2} \right) - \log_a \left( \frac{b^{1/3}}{c} \right)$.

    • $5 - \log_a c$
    • $5 + \log_a c$
    • $\log_a (a^5 c)$
    • $5 \log_a a - \log_a c^3$

    Answer: $5 - \log_a c$

  8. Given the system of equations:\n1. $\log_2 x + \log_2 y = 5$\n2. $\log_2 x - \log_2 y = 1$\nFind the values of $x$ and $y$.

    • $x=8, y=4$
    • $x=4, y=8$
    • $x=32, y=2$
    • $x=16, y=2$

    Answer: $x=8, y=4$

  9. Given $\log_b a = 4$, evaluate $\log_{b^2} (a^3)$.

    • $6$
    • $12$
    • $24$
    • $\frac{3}{8}$

    Answer: $6$

  10. If $\log_2 3 = A$, what is the value of $\log_4 9 + \log_8 27$? Round your answer to 3 decimal places.

    • $3.170$
    • $1.585$
    • $2.512$
    • $4.755$

    Answer: $3.170$

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