Solve 71=910(0.895)^3x

$$71 = 910 ( 0.895 ) ^ { 3 x }$$

Solve for x

$$x = \frac{\log_{\frac{200}{179}} {(\frac{910}{71})}}{3} \approx 7.664679935$$

Steps Using Definition of Logarithm

Divide both sides by 910.

$$\frac{71}{910}=0.895^{3x}$$

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

$$0.895^{3x}=\frac{71}{910}$$

Take the logarithm of both sides of the equation.

$$\log(0.895^{3x})=\log(\frac{71}{910})$$

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

$$3x\log(0.895)=\log(\frac{71}{910})$$

Divide both sides by \log(0.895).

$$3x=\frac{\log(\frac{71}{910})}{\log(0.895)}$$

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

$$3x=\log_{0.895}\left(\frac{71}{910}\right)$$

Divide both sides by 3.

$$x=\frac{\ln(\frac{71}{910})}{3\ln(\frac{179}{200})}$$

Solve for x (complex solution)

$$x=\frac{i\times 2\pi n_{1}}{3\ln(0.895)}+\frac{\log_{0.895}\left(\frac{71}{910}\right)}{3}$$

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