Solve e^(x/5)-7=9

$$e ^ { ( x / 5 ) } - 7 = 9$$

Solve for x

$$x=20\ln(2)\approx 13.862943611$$

Steps Using Definition of Logarithm

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

$$e^{\frac{1}{5}x}-7=9$$

Add 7 to both sides of the equation.

$$e^{\frac{1}{5}x}=16$$

Take the logarithm of both sides of the equation.

$$\log(e^{\frac{1}{5}x})=\log(16)$$

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

$$\frac{1}{5}x\log(e)=\log(16)$$

Divide both sides by \log(e).

$$\frac{1}{5}x=\frac{\log(16)}{\log(e)}$$

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

$$\frac{1}{5}x=\log_{e}\left(16\right)$$

Multiply both sides by 5.

$$x=\frac{4\ln(2)}{\frac{1}{5}}$$

Solve for x (complex solution)

$$x=20\ln(2)+i\times 10\pi n_{1}$$

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