Often, we will need to use the product property of logarithms, the quotient property of logarithms, and the power property of logarithms. We will use these properties to expand and condense logarithmic expressions.

Test Objectives
• Demonstrate an understanding of the properties of logarithms
• Demonstrate the ability to expand a logarithmic expression
• Demonstrate the ability to condense a logarithmic expression
Properties of Logarithms Practice Test:

#1:

Instructions: expand each. Assume all variables represent positive real numbers.

$$a)\hspace{.2em}\text{log}_{5}\left(\frac{x^4}{y^3}\right)$$

$$b)\hspace{.2em}\text{log}_{2}\left(12^3 \cdot 7^4\right)$$

#2:

Instructions: expand each. Assume all variables represent positive real numbers.

$$a)\hspace{.2em}\text{log}_{7}\left(z^2 \cdot \sqrt{x}\right)$$

$$b)\hspace{.2em}\text{log}_{5}\left(xy^3\right)^5$$

#3:

Instructions: condense each. Assume all variables represent positive real numbers.

$$a)\hspace{.2em}12\text{log}_{4}(a) + 4\text{log}_{4}(b) + 4\text{log}_{4}(c)$$

$$b)\hspace{.2em}\text{log}_{5}(x) + \text{log}_{5}(y) + \text{log}_{5}(w) + 2\text{log}_{5}(z)$$ $$b)\hspace{.2em}\text{log}_{5}(x) + \text{log}_{5}(y) + \text{log}_{5}(w)$$ $$+ \hspace{.2em}2\text{log}_{5}(z)$$

#4:

Instructions: condense each. Assume all variables represent positive real numbers.

$$a)\hspace{.2em}\text{log}_{4}(12) + 3\text{log}_{4}(5) + \frac{\text{log}_{4}(7)}{2}$$

$$b)\hspace{.2em}6\text{log}_{6}(u) - \text{log}_{6}(w) - 2\text{log}_{6}(v)$$

#5:

Instructions: condense each. Assume all variables represent positive real numbers.

$$a)\hspace{.2em}5\text{log}_{2}(x) + 5\text{log}_{2}(z) - 20\text{log}_{2}(y)$$

$$b)\hspace{.2em}4\text{log}_{4}(x) + 4\text{log}_{4}(z) - 20\text{log}_{4}(y)$$

Written Solutions:

#1:

Solutions:

$$a)\hspace{.2em}4\text{log}_{5}(x) - 3\text{log}_{5}(y)$$

$$b)\hspace{.2em}3\text{log}_{2}(12) + 4\text{log}_{2}(7)$$ Alternative answer: $$6 + 3\text{log}_2(3) + 4\text{log}_2(7)$$

#2:

Solutions:

$$a)\hspace{.2em}2\text{log}_{7}(z) + \frac{\text{log}_{7}(x)}{2}$$

$$b)\hspace{.2em}5\text{log}_{5}(x) + 15\text{log}_{5}(y)$$

#3:

Solutions:

$$a)\hspace{.2em}\text{log}_{4}(a^{12}b^4c^4)$$

$$b)\hspace{.2em}\text{log}_{5}(wyxz^2)$$

#4:

Solutions:

$$a)\hspace{.2em}\text{log}_{4}(12 \cdot 5^3 \sqrt{7})$$

$$b)\hspace{.2em}\text{log}_{6}\left(\frac{u^6}{wv^2}\right)$$

#5:

Solutions:

$$a)\hspace{.2em}\text{log}_{2}\left(\frac{x^5z^5}{y^{20}}\right)$$

$$b)\hspace{.2em}\text{log}_{4}\left(\frac{x^4z^4}{y^{20}}\right)$$