About Properties of Logarithms:
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
#1:
Instructions: Expand each. All variables are positive real numbers.
a) $$\log_{7}(2 \cdot 11\sqrt[3]{3 \cdot 11})$$
b) $$\log_{6}\left(\frac{x^6}{zy^6}\right)$$
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#2:
Instructions: Expand each.
a) $$\log_{6}\left(\frac{3}{8 \cdot 11^4}\right)^4$$
b) $$\log_{5}\sqrt{3 \cdot 5 \cdot 7 \cdot 16}$$
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#3:
Instructions: Condense each to a single logarithm. All variables are positive real numbers.
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) $$5\log_{2}(u) \hspace{.25em}- \hspace{.25em}5\log_{2}(w) \hspace{.25em}- \hspace{.25em}30\log_{2}(v)$$ $$5\log_{2}(u) \hspace{.25em}- \hspace{.25em}5\log_{2}(w) \hspace{.25em}- $$$$30\log_{2}(v)$$
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#4:
Instructions: Condense each to a single logarithm. All variables are positive real numbers.
a) $$18\log_{6}(c) \hspace{.25em}+ \hspace{.25em}18\log_{6}(a) \hspace{.25em}- \hspace{.25em}6\log_{6}(b)$$ $$18\log_{6}(c) \hspace{.25em}+ \hspace{.25em}18\log_{6}(a) \hspace{.25em}- $$$$6\log_{6}(b)$$
b) $$6\log_{10}(x) \hspace{.25em}+ \hspace{.25em}5\log_{10}(y) \hspace{.25em}+ \hspace{.25em}\log_{10}(z)$$ $$6\log_{10}(x) \hspace{.25em}+ \hspace{.25em}5\log_{10}(y) \hspace{.25em}+ $$$$\log_{10}(z)$$
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#5:
Instructions: Condense each to a single logarithm. All variables are positive real numbers.
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) $$10\log_{6}(x) \hspace{.25em}- \hspace{.25em}2\log_{6}(y) \hspace{.25em}- \hspace{.25em}2\log_{6}(z)$$ $$10\log_{6}(x) \hspace{.25em}- \hspace{.25em}2\log_{6}(y) \hspace{.25em}- $$$$2\log_{6}(z)$$
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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) $$6\log_{6}(x)\hspace{.25em}- \hspace{.25em}\log_{6}(z) \hspace{.25em}- \hspace{.25em}6\log_{6}(y)$$ $$6\log_{6}(x)\hspace{.25em}- \hspace{.25em}\log_{6}(z) \hspace{.25em}- $$$$6\log_{6}(y)$$
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#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) $$\frac{1}{2}\log_{5}(3) \hspace{.25em}+ \hspace{.25em}\frac{1}{2}\hspace{.25em}+ \hspace{.25em}\frac{1}{2}\log_{5}(7) \hspace{.25em}+ \hspace{.25em}\log_{5}(4) $$ $$\frac{1}{2}\log_{5}(3) \hspace{.25em}+ \hspace{.25em}\frac{1}{2}\hspace{.25em}+ $$$$ \frac{1}{2}\log_{5}(7) \hspace{.25em}+ \hspace{.25em}\log_{5}(4) $$
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#3:
Solutions:
a) $$\log_{10}(w^3\sqrt{uv})$$
b) $$\log_{2}\left(\frac{u^5}{w^5v^{30}}\right)$$
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#4:
Solutions:
a) $$\log_{6}\left(\frac{c^{18}a^{18}}{b^6}\right)$$
b) $$\log_{10}(x^6y^5z)$$
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#5:
Solutions:
a) $$\log_{5}(20\sqrt{66})$$
b) $$\log_{6}\left(\frac{x^{10}}{y^2z^2}\right)$$
