Solve 7^7/7^3=7^x

$$\frac { 7 ^ { 7 } } { 7 ^ { 3 } } = 7 ^ { x }$$

Solve for x

$$x=4$$

Steps Using Definition of Logarithm

To divide powers of the same base, subtract the denominator's exponent from the numerator's exponent. Subtract 3 from 7 to get 4.

$$7^{4}=7^{x}$$

Calculate 7 to the power of 4 and get 2401.

$$2401=7^{x}$$

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

$$7^{x}=2401$$

Take the logarithm of both sides of the equation.

$$\log(7^{x})=\log(2401)$$

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

$$x\log(7)=\log(2401)$$

Divide both sides by \log(7).

$$x=\frac{\log(2401)}{\log(7)}$$

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

$$x=\log_{7}\left(2401\right)$$

Solve for x (complex solution)

$$x=\frac{2\pi n_{1}i}{\ln(7)}+4$$

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