Question

16:31 62 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) \( \quad 1331=(121)^{\frac{3}{2}} \) 2 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) \( \log _{16} 50=\frac{2-a}{4 a} \)

          16:31
62

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) \( \quad 1331=(121)^{\frac{3}{2}} \)

2

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) \( \log _{16} 50=\frac{2-a}{4 a} \)

16:31
62

Use laws of logarithms to expand each of the expressions below.
(a) log2 8 x
(b) log3(x √(y))
(c) log√(a^5)
(d) loga((c^2)/(9 t^3))
(e) log((3 x^2)/((x+1)^10))
(f) ln√(a^2+b^2)
(g) log3((2 x)/(y))

Question 4

Simplify each expression and eliminate any negative exponent.
(a) (x^-1+y^-1)/((x y)^-1)
(b) ((3 x^5 y^2)/(6 x^5 y^-2))^-4
(c) ((x y^-1 z)^-2)/(((x z^-2)/(2 y^2))^-3)
(d) ((b^2 a)/(c))^4((a)/(b))^3
(e) ((2 x^3 y^2)/(z^3))^2((x^4 x^2)/(4 y^5))
(f) ((x^4 z^2)/(4 y^5))((2 x^3 y^2)/(z^3))^2

Question 5

Evaluate the following expressions.
(a) log√(10)
(b) log9√(3)
(c) ln((1)/(e))
(d) log4 8
(e) 5^log5 25
(f) log4 16^10

Question 6

Use the definition of logarithms to find the value of x in
(a) logx 1000=3
(b) logx 6=(1)/(2)
(c) ln((1)/(e))=x
(d) ln((1)/(e^2))=x
(e) log4 4=x
(f) log10(1-x)=-1
(g) log10 x=-1
(h) logx 8=(3)/(2)

Question 7

Express the following in logarithm form:
(a) 5^3=125
(b) 8^(-1)/(3)=(1)/(2)
(c) 1331=(121)^(3)/(2)

2

Question 8

Express the following equation in exponential form:
(a) log3 5=x
(b) log7(3 y)=2

Question 9

If log10 2=a, prove that:
(a) log8 5=(1-a)/(3 a)
(b) log16 50=(2-a)/(4 a)

Added by Esli E.

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