Solve 120=800*{0.1}^frac{x{6}}

$$120=800 \times { 0.1 }^{ \frac{ x }{ 6 } }$$

Solve for x

$$x=-6\log_{10}\left(0.15\right)\approx 4.943452446$$

Steps Using Definition of Logarithm

Divide both sides by 800.

$$\frac{120}{800}=0.1^{\frac{x}{6}}$$

Reduce the fraction \frac{120}{800} to lowest terms by extracting and canceling out 40.

$$\frac{3}{20}=0.1^{\frac{x}{6}}$$

Swap sides so that all variable terms are on the left hand side.

$$0.1^{\frac{x}{6}}=\frac{3}{20}$$

Use the rules of exponents and logarithms to solve the equation.

$$0.1^{\frac{1}{6}x}=\frac{3}{20}$$

Take the logarithm of both sides of the equation.

$$\log(0.1^{\frac{1}{6}x})=\log(\frac{3}{20})$$

The logarithm of a number raised to a power is the power times the logarithm of the number.

$$\frac{1}{6}x\log(0.1)=\log(\frac{3}{20})$$

Divide both sides by \log(0.1).

$$\frac{1}{6}x=\frac{\log(\frac{3}{20})}{\log(0.1)}$$

By the change-of-base formula \frac{\log(a)}{\log(b)}=\log_{b}\left(a\right).

$$\frac{1}{6}x=\log_{0.1}\left(\frac{3}{20}\right)$$

Multiply both sides by 6.

$$x=-\frac{\log(\frac{3}{20})}{\frac{1}{6}}$$

Solve for x (complex solution)

$$x=-\frac{i\times 12\pi n_{1}}{\ln(10)}-6\log_{10}\left(0.15\right)$$

$$n_{1}\in \mathrm{Z}$$