Solve 5^2*left(5^3right)^x=5^14

$${ 5 }^{ 2 } \times { \left( { 5 }^{ 3 } \right) }^{ x } = { 5 }^{ 14 }$$

Solve for x

$$x=4$$

Steps Using Definition of Logarithm

Calculate 5 to the power of 2 and get 25.

$$25\times \left(5^{3}\right)^{x}=5^{14}$$

Calculate 5 to the power of 3 and get 125.

$$25\times 125^{x}=5^{14}$$

Calculate 5 to the power of 14 and get 6103515625.

$$25\times 125^{x}=6103515625$$

Divide both sides by 25.

$$125^{x}=244140625$$

Take the logarithm of both sides of the equation.

$$\log(125^{x})=\log(244140625)$$

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

$$x\log(125)=\log(244140625)$$

Divide both sides by \log(125).

$$x=\frac{\log(244140625)}{\log(125)}$$

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

$$x=\log_{125}\left(244140625\right)$$

Solve for x (complex solution)

$$x=\frac{2\pi n_{1}i}{3\ln(5)}+4$$

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