Solve e^5-3x=10

$$e ^ { 5 - 3 x } = 10$$

Solve for x

$$x=\frac{5-\ln(10)}{3}\approx 0.899138302$$

Steps Using Definition of Logarithm

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

$$e^{-3x+5}=10$$

Take the logarithm of both sides of the equation.

$$\log(e^{-3x+5})=\log(10)$$

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

$$\left(-3x+5\right)\log(e)=\log(10)$$

Divide both sides by \log(e).

$$-3x+5=\frac{\log(10)}{\log(e)}$$

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

$$-3x+5=\log_{e}\left(10\right)$$

Subtract 5 from both sides of the equation.

$$-3x=\ln(10)-5$$

Divide both sides by -3.

$$x=\frac{\ln(10)-5}{-3}$$

Solve for x (complex solution)

$$x=-\frac{i\times 2\pi n_{1}}{3}-\frac{\ln(10)}{3}+\frac{5}{3}$$

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