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Use laws of logarithms to expand each of the expressions below.
(a) \( \log _{2} 8 x \)
(b) \( \quad \log _{3}(x \sqrt{y}) \)
(c) \( \log \sqrt{a^{5}} \)
(d) \( \quad \log _{a}\left(\frac{c^{2}}{9 t^{3}}\right) \)
(e) \( \quad \log \left(\frac{3 x^{2}}{(x+1)^{10}}\right) \)
(f) \( \ln \sqrt[4]{a^{2}+b^{2}} \)
(g) \( \log _{3}\left(\frac{2 x}{y}\right) \)
Question 4
Simplify each expression and eliminate any negative exponent.
(a) \( \frac{x^{-1}+y^{-1}}{(x y)^{-1}} \)
(b) \( \left(\frac{3 x^{5} y^{2}}{6 x^{5} y^{-2}}\right)^{-4} \)
(c) \( \frac{\left(x y^{-1} z\right)^{-2}}{\left(\frac{x z^{-2}}{2 y^{2}}\right)^{-3}} \)
(d) \( \left(\frac{b^{2} a}{c}\right)^{4}\left(\frac{a}{b}\right)^{3} \)
(e) \( \left(\frac{2 x^{3} y^{2}}{z^{3}}\right)^{2}\left(\frac{x^{4} x^{2}}{4 y^{5}}\right) \)
(f) \( \left(\frac{x^{4} z^{2}}{4 y^{5}}\right)\left(\frac{2 x^{3} y^{2}}{z^{3}}\right)^{2} \)
Question 5
Evaluate the following expressions.
(a) \( \log \sqrt{10} \)
(b) \( \log _{9} \sqrt{3} \)
(c) \( \ln \left(\frac{1}{e}\right) \)
(d) \( \log _{4} 8 \)
(e) \( 5^{\log _{5} 25} \)
(f) \( \log _{4} 16^{10} \)
Question 6
Use the definition of logarithms to find the value of \( x \) in
(a) \( \log _{x} 1000=3 \)
(b) \( \log _{x} 6=\frac{1}{2} \)
(c) \( \ln \left(\frac{1}{e}\right)=x \)
(d) \( \ln \left(\frac{1}{e^{2}}\right)=x \)
(e) \( \log _{4} 4=x \)
(f) \( \log _{10}(1-x)=-1 \)
(g) \( \log _{10} x=-1 \)
(h) \( \log _{x} 8=\frac{3}{2} \)
Question 7
Express the following in logarithm form:
(a) \( 5^{3}=125 \)
(b) \( 8^{\frac{-1}{3}}=\frac{1}{2} \)
(c) \( \quad 1331=(121)^{\frac{3}{2}} \)
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Question 8
Express the following equation in exponential form:
(a) \( \quad \log _{3} 5=x \)
(b) \( \quad \log _{7}(3 y)=2 \)
Question 9
If \( \log _{10} 2=a \), prove that:
(a) \( \quad \log _{8} 5=\frac{1-a}{3 a} \)
(b) \( \log _{16} 50=\frac{2-a}{4 a} \)
