Solve 33e^(33x)-69=0

$$33 e ^ { ( 33 x ) } - 69 = 0$$

Solve for x

$$x=\frac{\ln(\frac{23}{11})}{33}\approx 0.022351483$$

Steps Using Definition of Logarithm

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

$$33e^{33x}-69=0$$

Add 69 to both sides of the equation.

$$33e^{33x}=69$$

Divide both sides by 33.

$$e^{33x}=\frac{23}{11}$$

Take the logarithm of both sides of the equation.

$$\log(e^{33x})=\log(\frac{23}{11})$$

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

$$33x\log(e)=\log(\frac{23}{11})$$

Divide both sides by \log(e).

$$33x=\frac{\log(\frac{23}{11})}{\log(e)}$$

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

$$33x=\log_{e}\left(\frac{23}{11}\right)$$

Divide both sides by 33.

$$x=\frac{\ln(\frac{23}{11})}{33}$$

Solve for x (complex solution)

$$x=\frac{2\pi n_{1}i}{33}+\frac{\ln(\frac{23}{11})}{33}$$

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