# How do you solve 8^x=1000?

Nov 28, 2016

$x \approx 3.322$

#### Explanation:

Convert the exponential form to a logarithmic form ${a}^{x} = b \to x = {\log}_{a} b$

${8}^{x} = 1000$

$x = {\log}_{8} 1000$

You can use the 'change of base law' to calculate it.

${\log}_{a} b = \frac{{\log}_{c} b}{{\log}_{c} a} \text{ }$ (c is usually 10)

$x = {\log}_{10} 1000 / {\log}_{10} 8$

$x \approx 3.322$

Nov 29, 2016

x≈3.322

#### Explanation:

Use the $\textcolor{b l u e}{\text{law of logarithms}}$

$\textcolor{red}{\overline{\underline{| \textcolor{w h i t e}{\frac{2}{2}} \textcolor{b l a c k}{\log {x}^{n} = n \log x} \textcolor{w h i t e}{\frac{2}{2}} |}}}$
Applies to logarithms to any base.

Take the ln ( natural log) of both sides.

$\Rightarrow \ln {8}^{x} = \ln 1000$

Using the above law.

$\Rightarrow x \ln 8 = \ln 1000$

divide both sides by ln8

$\frac{x \cancel{\ln 8}}{\cancel{\ln 8}} = \ln \frac{1000}{\ln} 8$

rArrx≈3.322" to 3 decimal places"