Logarithms — Practice Quiz

A Algebra cheat sheet for Logarithms — every key formula with its symbols defined — plus a medium-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 $\log_3(9x)$.

    • $\log_3 9 + \log_3 x$
    • $2 + \log_3 x$
    • $2\log_3 x$
    • $9\log_3 x$

    Answer: $2 + \log_3 x$

  2. Expand $\log_2(\frac{8}{y})$.

    • $\log_2 8 - \log_2 y$
    • $3 - \log_2 y$
    • $\frac{\log_2 8}{\log_2 y}$
    • $3\log_2 y$

    Answer: $3 - \log_2 y$

  3. Simplify $\log_5(25^3)$.

    • $3\log_5 25$
    • $3 \times 2$
    • $6$
    • $25^3$

    Answer: $6$

  4. Express $2\log x + \log y$ as a single logarithm.

    • $\log(x^2 y)$
    • $\log(x^2 + y)$
    • $\log(2xy)$
    • $\log((xy)^2)$

    Answer: $\log(x^2 y)$

  5. Simplify $\log_b(\frac{x^4}{y^2})$.

    • $4\log_b x - 2\log_b y$
    • $\frac{4\log_b x}{2\log_b y}$
    • $\log_b(x^4) - \log_b(y^2)$
    • $2\log_b(\frac{x^2}{y})$

    Answer: $4\log_b x - 2\log_b y$

  6. If $\log_2 5 = A$, what is $\log_4 5$ in terms of $A$?

    • $\frac{A}{2}$
    • $2A$
    • $A^2$
    • $\frac{1}{A}$

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

  7. Evaluate $7^{\log_7 12}$.

    • $12$
    • $7$
    • $\log_7 12$
    • $1$

    Answer: $12$

  8. If $\log_3 x = 4$, what is $x$?

    • $3^4$
    • $81$
    • $4^3$
    • $64$

    Answer: $81$

  9. Simplify $\log_2(\frac{4x^3}{y})$.

    • $2 + 3\log_2 x - \log_2 y$
    • $2\log_2 x - \log_2 y$
    • $\log_2 4 + \log_2 x^3 - \log_2 y$
    • $2 + \log_2 x^3 - \log_2 y$

    Answer: $2 + 3\log_2 x - \log_2 y$

  10. Evaluate $\log_9 27$.

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

    Answer: $\frac{3}{2}$

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