Solve 5^3+4^5=5^n+1-1/1

$$5 ^ { 3 } + 4 ^ { 5 } = \frac { 5 ^ { n + 1 } - 1 } { 1 }$$

Solve for n

$$n=\log_{5}\left(46\right)+1\approx 3.378868652$$

Steps Using Definition of Logarithm

Calculate 5 to the power of 3 and get 125.

$$125+4^{5}=\frac{5^{n+1}-1}{1}$$

Calculate 4 to the power of 5 and get 1024.

$$125+1024=\frac{5^{n+1}-1}{1}$$

Add 125 and 1024 to get 1149.

$$1149=\frac{5^{n+1}-1}{1}$$

Anything divided by one gives itself.

$$1149=5^{n+1}-1$$

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

$$5^{n+1}-1=1149$$

Add 1 to both sides of the equation.

$$5^{n+1}=1150$$

Take the logarithm of both sides of the equation.

$$\log(5^{n+1})=\log(1150)$$

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

$$\left(n+1\right)\log(5)=\log(1150)$$

Divide both sides by \log(5).

$$n+1=\frac{\log(1150)}{\log(5)}$$

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

$$n+1=\log_{5}\left(1150\right)$$

Subtract 1 from both sides of the equation.

$$n=\log_{5}\left(1150\right)-1$$