Solve 5^2(3^0+6^2x)=2525/36

$$5 ^ { 2 } ( 3 ^ { 0 } + 6 ^ { 2 x } ) = 25 \frac { 25 } { 36 }$$

Solve for x

$$x=-1$$

Steps Using Definition of Logarithm

Multiply both sides of the equation by 36.

$$36\times 5^{2}\left(3^{0}+6^{2x}\right)=25\times 36+25$$

Calculate 5 to the power of 2 and get 25.

$$36\times 25\left(3^{0}+6^{2x}\right)=25\times 36+25$$

Multiply 36 and 25 to get 900.

$$900\left(3^{0}+6^{2x}\right)=25\times 36+25$$

Calculate 3 to the power of 0 and get 1.

$$900\left(1+6^{2x}\right)=25\times 36+25$$

Use the distributive property to multiply 900 by 1+6^{2x}.

$$900+900\times 6^{2x}=25\times 36+25$$

Multiply 25 and 36 to get 900.

$$900+900\times 6^{2x}=900+25$$

Add 900 and 25 to get 925.

$$900+900\times 6^{2x}=925$$

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

$$900\times 6^{2x}+900=925$$

Subtract 900 from both sides of the equation.

$$900\times 6^{2x}=25$$

Divide both sides by 900.

$$6^{2x}=\frac{1}{36}$$

Take the logarithm of both sides of the equation.

$$\log(6^{2x})=\log(\frac{1}{36})$$

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

$$2x\log(6)=\log(\frac{1}{36})$$

Divide both sides by \log(6).

$$2x=\frac{\log(\frac{1}{36})}{\log(6)}$$

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

$$2x=\log_{6}\left(\frac{1}{36}\right)$$

Divide both sides by 2.

$$x=-\frac{2}{2}$$

Solve for x (complex solution)

$$x=\frac{\pi n_{1}i}{\ln(6)}-1$$

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