Solve [(5^3)^2div25]^x=1

$$[ ( 5 ^ { 3 } ) ^ { 2 } \div 25 ] ^ { x } = 1$$

Solve for x

$$x=0$$

Steps Using Definition of Logarithm

To raise a power to another power, multiply the exponents. Multiply 3 and 2 to get 6.

$$\left(\frac{5^{6}}{25}\right)^{x}=1$$

Calculate 5 to the power of 6 and get 15625.

$$\left(\frac{15625}{25}\right)^{x}=1$$

Divide 15625 by 25 to get 625.

$$625^{x}=1$$

Take the logarithm of both sides of the equation.

$$\log(625^{x})=\log(1)$$

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

$$x\log(625)=\log(1)$$

Divide both sides by \log(625).

$$x=\frac{\log(1)}{\log(625)}$$

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

$$x=\log_{625}\left(1\right)$$

Solve for x (complex solution)

$$x=\frac{\pi n_{1}i}{2\ln(5)}$$

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