Expand Logarithms Worksheets (printable, online, answers, examples) - Free Printable

Explanation:
We are asked to expand each logarithm using logarithmic rules. The main rules we’ll use are:

1. Power Rule:
$\log_b (x^k) = k \cdot \log_b x$

2. Quotient Rule:
$\log_b \left(\frac{x}{y}\right) = \log_b x - \log_b y$

3. Product Rule:
$\log_b (xy) = \log_b x + \log_b y$

Let’s go one by one.

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1. $\log_5 \left( \frac{m}{s} \right)^4$

First, apply the power rule:
$\log_5 \left( \frac{m}{s} \right)^4 = 4 \cdot \log_5 \left( \frac{m}{s} \right)$

Then apply the quotient rule inside:
$= 4 \cdot \left( \log_5 m - \log_5 s \right)$

Distribute the 4:
$= 4\log_5 m - 4\log_5 s$

✔ Done.

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2. $\log_2 \left( \frac{n^5}{w^4} \right)$

Apply quotient rule first:
$= \log_2 (n^5) - \log_2 (w^4)$

Now apply power rule to each term:
$= 5\log_2 n - 4\log_2 w$

✔ Done.

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3. $\log_8 \left( a^3 b \right)^2$

First, apply power rule to the whole expression:
$= 2 \cdot \log_8 (a^3 b)$

Now apply product rule inside:
$= 2 \cdot \left( \log_8 (a^3) + \log_8 b \right)$

Apply power rule to $a^3$:
$= 2 \cdot \left( 3\log_8 a + \log_8 b \right)$

Distribute the 2:
$= 6\log_8 a + 2\log_8 b$

✔ Done.

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4. $\log_4 \left( \frac{p^3 m}{n^2} \right)$

Apply quotient rule:
$= \log_4 (p^3 m) - \log_4 (n^2)$

Apply product rule to numerator:
$= \left( \log_4 p^3 + \log_4 m \right) - \log_4 n^2$

Apply power rule to $p^3$ and $n^2$:
$= (3\log_4 p + \log_4 m) - 2\log_4 n$

Remove parentheses:
$= 3\log_4 p + \log_4 m - 2\log_4 n$

✔ Done.

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5. $\log_9 \left( \frac{w^2 t^3}{r s^2} \right)$

Apply quotient rule:
$= \log_9 (w^2 t^3) - \log_9 (r s^2)$

Apply product rule to both numerator and denominator:
$= \left( \log_9 w^2 + \log_9 t^3 \right) - \left( \log_9 r + \log_9 s^2 \right)$

Apply power rule:
$= (2\log_9 w + 3\log_9 t) - (\log_9 r + 2\log_9 s)$

Distribute the minus sign:
$= 2\log_9 w + 3\log_9 t - \log_9 r - 2\log_9 s$

✔ Done.

Final Answer:
1. $4\log_5 m - 4\log_5 s$
2. $5\log_2 n - 4\log_2 w$
3. $6\log_8 a + 2\log_8 b$
4. $3\log_4 p + \log_4 m - 2\log_4 n$
5. $2\log_9 w + 3\log_9 t - \log_9 r - 2\log_9 s$