Solve 5^a+1+5^2-9=126

$$5 ^ { a + 1 } + 5 ^ { 2 - 9 } = 126$$

Solve for a

$$a=\log_{5}\left(9843749\right)-8\approx 2.004950839$$

Steps Using Definition of Logarithm

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

$$5^{a+1}+\frac{1}{78125}=126$$

Subtract \frac{1}{78125} from both sides of the equation.

$$5^{a+1}=\frac{9843749}{78125}$$

Take the logarithm of both sides of the equation.

$$\log(5^{a+1})=\log(\frac{9843749}{78125})$$

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

$$\left(a+1\right)\log(5)=\log(\frac{9843749}{78125})$$

Divide both sides by \log(5).

$$a+1=\frac{\log(\frac{9843749}{78125})}{\log(5)}$$

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

$$a+1=\log_{5}\left(\frac{9843749}{78125}\right)$$

Subtract 1 from both sides of the equation.

$$a=\log_{5}\left(9843749\right)-7-1$$