Solve 75=7^4x

$$75 = 7 ^ { 4 x }$$

Solve for x

$$x=\frac{\log_{7}\left(75\right)}{4}\approx 0.554687496$$

Steps Using Definition of Logarithm

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

$$7^{4x}=75$$

Take the logarithm of both sides of the equation.

$$\log(7^{4x})=\log(75)$$

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

$$4x\log(7)=\log(75)$$

Divide both sides by \log(7).

$$4x=\frac{\log(75)}{\log(7)}$$

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

$$4x=\log_{7}\left(75\right)$$

Divide both sides by 4.

$$x=\frac{\log_{7}\left(75\right)}{4}$$

Solve for x (complex solution)

$$x=\frac{\pi n_{1}i}{2\ln(7)}+\frac{\log_{7}\left(75\right)}{4}$$

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