Question

question 6
evaluate the following logarithms without a calculator (no decimal answers):
\\(\log \left(10^{-8}\
ight) = \square\\)
\\(\log \left(\sqrt4{1000}\
ight) = \square\\)
\\(\ln \left(e^{-5}\
ight) = \square\\)
\\(\ln \left(\sqrt3{e^{2}}\
ight) = \square\\)
question help: video

Explanation:

Step1: Evaluate $\log(10^{-8})$

Recall the logarithm property $\log(a^b)=b\log(a)$. For $\log(10^{-8})$, since $\log(10) = 1$ (base 10 logarithm), we have $-8\log(10)=-8\times1=-8$.

Step2: Evaluate $\log(\sqrt[4]{1000})$

First, rewrite $\sqrt[4]{1000}$ as $1000^{\frac{1}{4}}$. And $1000 = 10^3$, so $1000^{\frac{1}{4}}=(10^3)^{\frac{1}{4}} = 10^{\frac{3}{4}}$. Then $\log(10^{\frac{3}{4}})=\frac{3}{4}\log(10)=\frac{3}{4}\times1=\frac{3}{4}$.

Step3: Evaluate $\ln(e^{-5})$

Recall the natural logarithm property $\ln(e^b)=b$. So $\ln(e^{-5})=-5$.

Step4: Evaluate $\ln(\sqrt[3]{e^2})$

Rewrite $\sqrt[3]{e^2}$ as $e^{\frac{2}{3}}$. Then $\ln(e^{\frac{2}{3}})=\frac{2}{3}$.

Answer:

$\log(10^{-8})=-8$, $\log(\sqrt[4]{1000})=\frac{3}{4}$, $\ln(e^{-5})=-5$, $\ln(\sqrt[3]{e^2})=\frac{2}{3}$