Solve 8000=6500e^.06x

$$8000=6500 { e }^{ .06x }$$

Solve for x

$$x=\frac{200\ln(2)-50\ln(13)}{3}\approx 3.46065608$$

Steps Using Definition of Logarithm

Divide both sides by 6500.

$$\frac{8000}{6500}=e^{0.06x}$$

Reduce the fraction \frac{8000}{6500} to lowest terms by extracting and canceling out 500.

$$\frac{16}{13}=e^{0.06x}$$

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

$$e^{0.06x}=\frac{16}{13}$$

Take the logarithm of both sides of the equation.

$$\log(e^{0.06x})=\log(\frac{16}{13})$$

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

$$0.06x\log(e)=\log(\frac{16}{13})$$

Divide both sides by \log(e).

$$0.06x=\frac{\log(\frac{16}{13})}{\log(e)}$$

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

$$0.06x=\log_{e}\left(\frac{16}{13}\right)$$

Divide both sides of the equation by 0.06, which is the same as multiplying both sides by the reciprocal of the fraction.

$$x=\frac{\ln(\frac{16}{13})}{0.06}$$

Solve for x (complex solution)

$$x=\frac{i\times 100\pi n_{1}}{3}+\frac{200\ln(2)}{3}-\frac{50\ln(13)}{3}$$

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