Chapter 6 - Exponential, Logarithmic, And Inverse Trigonometric Functions - 6.1 Exponential And Logarithmic Functions - Exercises Set 6.1 - Page 418: 6

$$\left( {\bf{a}} \right) - 3,\,\,\,\left( {\bf{b}} \right)4,\,\,\,\left( {\bf{c}} \right)3,\,\,\,\left( {\bf{d}} \right)\frac{1}{2}$$

$$\eqalign{ & \left( {\bf{a}} \right){\log _{10}}\left( {0.001} \right) \cr & {\text{write }}0.001{\text{ as }}\frac{1}{{{{10}^3}}} \cr & = {\log _{10}}\left( {\frac{1}{{{{10}^3}}}} \right) \cr & {\text{use the exponential property }}\frac{1}{{{a^n}}} = {a^{ - n}} \cr & = {\log _{10}}\left( {{{10}^{ - 3}}} \right) \cr & {\text{use the logarithmic property }}\log {a^n} = n\log a \cr & = - 3{\log _{10}}\left( {10} \right) \cr & {\text{simplify}} \cr & = - 3\left( 1 \right) \cr & = - 3 \cr & \cr & \left( {\bf{b}} \right){\log _{10}}\left( {{{10}^4}} \right) \cr & {\text{use the logarithmic property }}\log {a^n} = n\log a \cr & = 4{\log _{10}}\left( {10} \right) \cr & {\text{use the logarithmic property }}{\log _a}a = 1 \cr & = 4\left( 1 \right) \cr & = 4 \cr & \cr & \left( {\bf{c}} \right)\ln \left( {{e^3}} \right) \cr & {\text{use the logarithmic property }}\ln {a^n} = n\ln a \cr & = 3\ln e \cr & {\text{simplify}} \cr & = 3\left( 1 \right) \cr & = 3 \cr & \cr & \left( {\bf{d}} \right)\ln \left( {\sqrt e } \right) \cr & {\text{write }}\sqrt e {\text{ as }}{e^{1/2}} \cr & = \ln {e^{1/2}} \cr & {\text{use the logarithmic property }}\ln {a^n} = n\ln a \cr & = \frac{1}{2}\ln e \cr & {\text{simplify}} \cr & = \frac{1}{2}\left( 1 \right) \cr & = \frac{1}{2} \cr} $$