Solve e^0.19y=7

$$e ^ { 0.19 y } = 7$$

Solve for y

$$y=\frac{100\ln(7)}{19}\approx 10.241632363$$

Steps Using Definition of Logarithm

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

$$e^{0.19y}=7$$

Take the logarithm of both sides of the equation.

$$\log(e^{0.19y})=\log(7)$$

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

$$0.19y\log(e)=\log(7)$$

Divide both sides by \log(e).

$$0.19y=\frac{\log(7)}{\log(e)}$$

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

$$0.19y=\log_{e}\left(7\right)$$

Divide both sides of the equation by 0.19, which is the same as multiplying both sides by the reciprocal of the fraction.

$$y=\frac{\ln(7)}{0.19}$$

Solve for y (complex solution)

$$y=\frac{i\times 200\pi n_{1}}{19}+\frac{100\ln(7)}{19}$$

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