Part a)
For this part, we can use the following properties:
[tex]\begin{gathered} \frac{1}{a^n}=a^{-n}\Rightarrow\text{ Property of the exponents} \\ \log _aa^x=x\Rightarrow\text{ Property of logarithms} \end{gathered}[/tex]
So, applying the above property of exponents, we have:
[tex]\begin{gathered} \frac{1}{9}=\frac{1}{3\cdot3} \\ \frac{1}{9}=\frac{1}{3^2} \\ \frac{1}{9}=3^{-2} \end{gathered}[/tex]
Now, applying the above property of logarithms, we have:
[tex]\begin{gathered} \log _3\frac{1}{9}=\log _33^{-2} \\ $\boldsymbol{\log _3\frac{1}{9}=-2}$ \end{gathered}[/tex]Part b)
For this part, we can apply the following property of logarithms:
[tex]\log _a1=0[/tex]
Then, in this case, we have:
[tex]\begin{gathered} a=5 \\ \log _a1=0 \\ \boldsymbol{\log _51=0} \end{gathered}[/tex]Part c)
For this part, we can apply the following property of logarithms:
[tex]\ln e^x=x[/tex]
So, we have:
[tex]\begin{gathered} x=5 \\ \ln e^x=x \\ $\boldsymbol{\ln e}^{\boldsymbol{5}}\boldsymbol{=5}$ \end{gathered}[/tex]Part d)
For this part, we can rewrite 0.00001 like this:
[tex]\begin{gathered} 0.00001=\frac{0.00001}{1} \\ 0.00001=\frac{0.00001\cdot100,000}{1\cdot100,000} \\ 0.00001=\frac{1}{100,000} \\ 0.00001=\frac{1}{10\cdot10\cdot10\cdot10\cdot10} \\ 0.00001=\frac{1}{10^5} \\ 0.00001=10^{-5} \end{gathered}[/tex]
Now, applying the above property of logarithms, we have:
[tex]\begin{gathered} a=10\text{ and }x=-5 \\ \log _aa^x=x \\ \log 0.00001=\log _{10}10^{-5} \\ $\boldsymbol{\log 0.00001=-5}$ \end{gathered}[/tex]
