The product rule for logarithms allows us to write the logarithm of a product as the sum of the logarithms of the factors. The quotient rule for logarithms allows us to write the logarithm of a quotient as the difference between the logarithm of the numerator and the logarithm of the denominator. The power rule for logarithms tells us the logarithm of a number to a power is equal to the exponent multiplied by the logarithm of the number.

Test Objectives
• Demonstrate an understanding of the product rule, & quotient rule for logarithms
• Demonstrate an understanding of the power rule for logarithms
• Demonstrate the ability to expand & condense a logarithm
Properties of Logarithms Practice Test:

#1:

Instructions: Expand each.

a) $$\log_{7}(2 \cdot 11\sqrt[3]{3 \cdot 11})$$

b) $$\log_{6}\left(\frac{x^6}{zy^6}\right)$$

#2:

Instructions: Expand each.

a) $$\log_{6}\left(\frac{3}{8 \cdot 11^4}\right)^4$$

b) $$\log_{5}\sqrt{3 \cdot 5 \cdot 7 \cdot 16}$$

#3:

Instructions: Condense each to a single logarithm.

a) $$3\log_{10}(w) \hspace{.25em} + \hspace{.25em} \frac{1}{2}\log_{10}(u) \hspace{.25em} + \hspace{.25em} \frac{1}{2}\log_{10}(v)$$ $$3\log_{10}(w) \hspace{.25em} + \hspace{.25em} \frac{1}{2}\log_{10}(u) \hspace{.25em} +$$$$\frac{1}{2}\log_{10}(v)$$

b) $$5\log_{2}(u) \hspace{.25em} - \hspace{.25em} 5\log_{2}(w) \hspace{.25em} - \hspace{.25em} 30\log_{2}(v)$$ $$5\log_{2}(u) \hspace{.25em} - \hspace{.25em} 5\log_{2}(w) \hspace{.25em} -$$$$30\log_{2}(v)$$

#4:

Instructions: Condense each to a single logarithm.

a) $$18\log_{6}(c) \hspace{.25em} + \hspace{.25em} 18\log_{6}(a) \hspace{.25em} - \hspace{.25em} 6\log_{6}(b)$$ $$18\log_{6}(c) \hspace{.25em} + \hspace{.25em} 18\log_{6}(a) \hspace{.25em} -$$$$6\log_{6}(b)$$

a) $$6\log_{10}(x) \hspace{.25em} + \hspace{.25em} 5\log_{10}(y) \hspace{.25em} + \hspace{.25em} \log_{10}(z)$$ $$6\log_{10}(x) \hspace{.25em} + \hspace{.25em} 5\log_{10}(y) \hspace{.25em} +$$$$\log_{10}(z)$$

#5:

Instructions: Condense each to a single logarithm.

a) $$\log_{5}(10) \hspace{.25em} + \hspace{.25em} \frac{1}{2}\log_{5}(3) \hspace{.25em} + \hspace{.25em} \frac{1}{2}\log_{5}(8) \hspace{.25em} + \hspace{.25em} \frac{1}{2}\log_{5}(11)$$ $$\log_{5}(10) \hspace{.25em} + \hspace{.25em} \frac{1}{2}\log_{5}(3) \hspace{.25em} +$$$$\frac{1}{2}\log_{5}(8) \hspace{.25em} + \hspace{.25em} \frac{1}{2}\log_{5}(11)$$

b) $$10\log_{6}(x) \hspace{.25em} - \hspace{.25em} 2\log_{6}(y) \hspace{.25em} - \hspace{.25em} 2\log_{6}(z)$$ $$10\log_{6}(x) \hspace{.25em} - \hspace{.25em} 2\log_{6}(y) \hspace{.25em} -$$$$2\log_{6}(z)$$

Written Solutions:

#1:

Solutions:

a) $$\log_{7}(2)\hspace{.25em} + \hspace{.25em} \frac{4}{3}\log_{7}(11) \hspace{.25em} + \hspace{.25em} \frac{1}{3}\log_{7}(3)$$ $$\log_{7}(2)\hspace{.25em} + \hspace{.25em} \frac{4}{3}\log_{7}(11) \hspace{.25em} +$$$$\frac{1}{3}\log_{7}(3)$$

b) $$6\log_{6}(x)\hspace{.25em} - \hspace{.25em} \log_{6}(z) \hspace{.25em} - \hspace{.25em} 6\log_{6}(y)$$ $$6\log_{6}(x)\hspace{.25em} - \hspace{.25em} \log_{6}(z) \hspace{.25em} -$$$$6\log_{6}(y)$$

#2:

Solutions:

a) $$4\log_{6}(3) - 4\log_{6}(8) - 16\log_{6}(11)$$ $$4\log_{6}(3) - 4\log_{6}(8) -$$$$16\log_{6}(11)$$

b) $$\frac{1}{2}\log_{5}(3) \hspace{.25em} + \hspace{.25em} \frac{1}{2} \hspace{.25em} + \hspace{.25em} \frac{1}{2}\log_{5}(7) \hspace{.25em} + \hspace{.25em} \log_{5}(4)$$ $$\frac{1}{2}\log_{5}(3) \hspace{.25em} + \hspace{.25em} \frac{1}{2} \hspace{.25em} +$$$$\frac{1}{2}\log_{5}(7) \hspace{.25em} + \hspace{.25em} \log_{5}(4)$$

#3:

Solutions:

a) $$\log_{10}(w^3\sqrt{uv})$$

b) $$\log_{2}\left(\frac{u^5}{w^5v^{30}}\right)$$

#4:

Solutions:

a) $$\log_{6}\left(\frac{c^{18}a^{18}}{b^6}\right)$$

b) $$\log_{10}(x^6y^5z)$$

#5:

Solutions:

a) $$\log_{5}(20\sqrt{66})$$

b) $$\log_{6}\left(\frac{x^{10}}{y^2z^2}\right)$$