Properties of Logarithms - Kuta Software - Free Printable

Here are the step-by-step solutions to expand each logarithm.

To solve these, we use three main rules:
1. Product Rule: $\log(a \cdot b) = \log a + \log b$ (Multiplication inside becomes addition outside).
2. Quotient Rule: $\log(\frac{a}{b}) = \log a - \log b$ (Division inside becomes subtraction outside).
3. Power Rule: $\log(a^n) = n \cdot \log a$ (An exponent inside moves to the front as a multiplier).

1) $\log(6 \cdot 11)$
* Use the Product Rule to split the multiplication.
* $\log 6 + \log 11$

2) $\log(5 \cdot 3)$
* Use the Product Rule to split the multiplication.
* $\log 5 + \log 3$

3) $\log\left(\frac{6}{11}\right)^5$
* First, use the Power Rule to move the exponent $5$ to the front.
* $5 \log\left(\frac{6}{11}\right)$
* Next, use the Quotient Rule to split the fraction. Remember to keep the $5$ distributed to both parts.
* $5(\log 6 - \log 11)$
* Distribute the $5$.
* $5 \log 6 - 5 \log 11$

4) $\log(3 \cdot 2^3)$
* Use the Product Rule to separate the $3$ and the $2^3$.
* $\log 3 + \log(2^3)$
* Use the Power Rule on the second part to move the exponent $3$ to the front.
* $\log 3 + 3 \log 2$

5) $\log\frac{2^4}{5}$
* Use the Quotient Rule to split the fraction.
* $\log(2^4) - \log 5$
* Use the Power Rule on the first part to move the exponent $4$ to the front.
* $4 \log 2 - \log 5$

6) $\log\left(\frac{6}{5}\right)^6$
* Use the Power Rule to move the exponent $6$ to the front.
* $6 \log\left(\frac{6}{5}\right)$
* Use the Quotient Rule to split the fraction inside.
* $6(\log 6 - \log 5)$
* Distribute the $6$.
* $6 \log 6 - 6 \log 5$

7) $\log\frac{x}{y^6}$
* Use the Quotient Rule to split the fraction.
* $\log x - \log(y^6)$
* Use the Power Rule on the second part to move the exponent $6$ to the front.
* $\log x - 6 \log y$

8) $\log(a \cdot b)^2$
* Use the Power Rule to move the exponent $2$ to the front.
* $2 \log(a \cdot b)$
* Use the Product Rule to split the multiplication inside.
* $2(\log a + \log b)$
* Distribute the $2$.
* $2 \log a + 2 \log b$

9) $\log\frac{u^4}{v}$
* Use the Quotient Rule to split the fraction.
* $\log(u^4) - \log v$
* Use the Power Rule on the first part to move the exponent $4$ to the front.
* $4 \log u - \log v$

10) $\log\frac{x}{y^5}$
* Use the Quotient Rule to split the fraction.
* $\log x - \log(y^5)$
* Use the Power Rule on the second part to move the exponent $5$ to the front.
* $\log x - 5 \log y$

11) $\log\sqrt[4]{x \cdot y \cdot z}$
* Rewrite the radical (root) as a fractional exponent. The 4th root is the power of $\frac{1}{4}$.
* $\log(x \cdot y \cdot z)^{\frac{1}{4}}$
* Use the Power Rule to move the $\frac{1}{4}$ to the front.
* $\frac{1}{4} \log(x \cdot y \cdot z)$
* Use the Product Rule to split the three variables being multiplied.
* $\frac{1}{4} (\log x + \log y + \log z)$
* Distribute the $\frac{1}{4}$.
* $\frac{1}{4} \log x + \frac{1}{4} \log y + \frac{1}{4} \log z$

12) $\log(x \cdot y \cdot z^2)$
* Use the Product Rule to split all three terms. Note that $z^2$ stays together for now.
* $\log x + \log y + \log(z^2)$
* Use the Power Rule on the last term to move the exponent $2$ to the front.
* $\log x + \log y + 2 \log z$

Final Answer:
1) $\log 6 + \log 11$
2) $\log 5 + \log 3$
3) $5 \log 6 - 5 \log 11$
4) $\log 3 + 3 \log 2$
5) $4 \log 2 - \log 5$
6) $6 \log 6 - 6 \log 5$
7) $\log x - 6 \log y$
8) $2 \log a + 2 \log b$
9) $4 \log u - \log v$
10) $\log x - 5 \log y$
11) $\frac{1}{4} \log x + \frac{1}{4} \log y + \frac{1}{4} \log z$
12) $\log x + \log y + 2 \log z$