Solve 1.6=e^0.09x

$$1.6 = e ^ { 0.09 x }$$

Solve for x

$$x=\frac{100\ln(2)}{3}-\frac{100\ln(5)}{9}\approx 5.222262547$$

Steps Using Definition of Logarithm

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

$$e^{0.09x}=1.6$$

Take the logarithm of both sides of the equation.

$$\log(e^{0.09x})=\log(1.6)$$

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

$$0.09x\log(e)=\log(1.6)$$

Divide both sides by \log(e).

$$0.09x=\frac{\log(1.6)}{\log(e)}$$

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

$$0.09x=\log_{e}\left(1.6\right)$$

Divide both sides of the equation by 0.09, which is the same as multiplying both sides by the reciprocal of the fraction.

$$x=\frac{\ln(\frac{8}{5})}{0.09}$$

Solve for x (complex solution)

$$x=\frac{i\times 200\pi n_{1}}{9}+\frac{100\ln(2)}{3}-\frac{100\ln(5)}{9}$$

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