Solve 49^x-2=343

$$49 ^ { x - 2 } = 343$$

Solve for x

$$x = \frac{7}{2} = 3\frac{1}{2} = 3.5$$

Steps Using Definition of Logarithm

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

$$49^{x-2}=343$$

Take the logarithm of both sides of the equation.

$$\log(49^{x-2})=\log(343)$$

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

$$\left(x-2\right)\log(49)=\log(343)$$

Divide both sides by \log(49).

$$x-2=\frac{\log(343)}{\log(49)}$$

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

$$x-2=\log_{49}\left(343\right)$$

Add 2 to both sides of the equation.

$$x=\frac{3}{2}-\left(-2\right)$$

Solve for x (complex solution)

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

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