Solve 5^6+6^7=7^n

$$5 ^ { 6 } + 6 ^ { 7 } = 7 ^ { n }$$

Solve for n

$$n=\log_{7}\left(295561\right)\approx 6.473387546$$

Steps Using Definition of Logarithm

Calculate 5 to the power of 6 and get 15625.

$$15625+6^{7}=7^{n}$$

Calculate 6 to the power of 7 and get 279936.

$$15625+279936=7^{n}$$

Add 15625 and 279936 to get 295561.

$$295561=7^{n}$$

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

$$7^{n}=295561$$

Take the logarithm of both sides of the equation.

$$\log(7^{n})=\log(295561)$$

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

$$n\log(7)=\log(295561)$$

Divide both sides by \log(7).

$$n=\frac{\log(295561)}{\log(7)}$$

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

$$n=\log_{7}\left(295561\right)$$