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. If $P = \log_a x$ and $Q = \log_a y$, which of the following expressions correctly represents $\log_a(\frac{x^3}{a \sqrt{y}})$ in terms of $P$ and $Q$?

    • $3P - \frac{1}{2}Q - 1$
    • $3P - \frac{1}{2}Q$
    • $\frac{3P}{1 + \frac{1}{2}Q}$
    • $3P - 2Q - 1$

    Answer: $3P - \frac{1}{2}Q - 1$

  2. Given that $\log_2 3 = A$ and $\log_3 5 = B$, express $\log_{12} 45$ in terms of $A$ and $B$.

    • $\frac{2A + AB}{2 + A}$
    • $\frac{A + 2AB}{2 + A}$
    • $\frac{2 + AB}{A + 2}$
    • $\frac{A + B}{2A + 1}$

    Answer: $\frac{2A + AB}{2 + A}$

  3. Solve for $x$: $\log_3(x^2 - 8) - \log_3 x = 1$.

    • $x = 4$
    • $x = 2$
    • $x = 4$ or $x = -2$
    • No real solution

    Answer: $x = 4$

  4. Simplify the expression: $e^{\ln(x^2) + \ln(y) - \ln(z^3)}$.

    • $\frac{x^2 y}{z^3}$
    • $\frac{x^2 + y}{z^3}$
    • $x^2 y - z^3$
    • $e^{x^2 y / z^3}$

    Answer: $\frac{x^2 y}{z^3}$

  5. If $\log_b a = 3$ and $\log_c b = 2$, what is the value of $\log_a c$?

    • $\frac{1}{6}$
    • $6$
    • $\frac{2}{3}$
    • $\frac{3}{2}$

    Answer: $\frac{1}{6}$

  6. Given $\log_x 2 = A$ and $\log_y 4 = B$. If $x = y^2$, express $\log_x 8$ in terms of $A$ and $B$.

    • $3A$
    • $\frac{3}{2}B$
    • $\frac{3}{2}A$
    • $\frac{3}{4}B$

    Answer: $3A$

  7. If $2^{\log_2 x} + 3^{\log_3 y} = 10$ and $\log_5(x+y) = 2$, find the value of $xy$.

    • $24$
    • $25$
    • $100$
    • $50$

    Answer: $24$

  8. Solve for $x$: $\log_x 2 + \log_{x^2} 4 = 3$.

    • $\sqrt{2}$
    • $2$
    • $4$
    • $\frac{1}{2}$

    Answer: $\sqrt{2}$

  9. Which of the following is equivalent to $\frac{1}{\log_a(abc)} + \frac{1}{\log_b(abc)} + \frac{1}{\log_c(abc)}$?

    • $1$
    • $\log_{abc}(a+b+c)$
    • $\log_{abc}(abc)$
    • $\log_a b c$

    Answer: $1$

  10. If $\log_a x = 2$, $\log_b x = 3$, and $\log_c x = 6$, what is $\log_{abc} x$?

    • $1$
    • $\frac{1}{11}$
    • $\frac{1}{6}$
    • $6$

    Answer: $1$

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