Solve 76=7^x+2

$$76 = 7 ^ { x } + 2$$

Solve for x

$$x=\log_{7}\left(74\right)\approx 2.211851917$$

Steps Using Definition of Logarithm

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

$$7^{x}+2=76$$

Subtract 2 from both sides of the equation.

$$7^{x}=74$$

Take the logarithm of both sides of the equation.

$$\log(7^{x})=\log(74)$$

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

$$x\log(7)=\log(74)$$

Divide both sides by \log(7).

$$x=\frac{\log(74)}{\log(7)}$$

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

$$x=\log_{7}\left(74\right)$$

Solve for x (complex solution)

$$x=\frac{2\pi n_{1}i}{\ln(7)}+\log_{7}\left(74\right)$$

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