Solve 8(9+1)(9^2+1)(9^4+1)(9^8+1)=9^x-1

$$8 ( 9 + 1 ) ( 9 ^ { 2 } + 1 ) ( 9 ^ { 4 } + 1 ) ( 9 ^ { 8 } + 1 ) = 9 ^ { x } - 1$$

Solve for x

$$x=16$$

Steps Using Definition of Logarithm

Add 9 and 1 to get 10.

$$8\times 10\left(9^{2}+1\right)\left(9^{4}+1\right)\left(9^{8}+1\right)=9^{x}-1$$

Multiply 8 and 10 to get 80.

$$80\left(9^{2}+1\right)\left(9^{4}+1\right)\left(9^{8}+1\right)=9^{x}-1$$

Calculate 9 to the power of 2 and get 81.

$$80\left(81+1\right)\left(9^{4}+1\right)\left(9^{8}+1\right)=9^{x}-1$$

Add 81 and 1 to get 82.

$$80\times 82\left(9^{4}+1\right)\left(9^{8}+1\right)=9^{x}-1$$

Multiply 80 and 82 to get 6560.

$$6560\left(9^{4}+1\right)\left(9^{8}+1\right)=9^{x}-1$$

Calculate 9 to the power of 4 and get 6561.

$$6560\left(6561+1\right)\left(9^{8}+1\right)=9^{x}-1$$

Add 6561 and 1 to get 6562.

$$6560\times 6562\left(9^{8}+1\right)=9^{x}-1$$

Multiply 6560 and 6562 to get 43046720.

$$43046720\left(9^{8}+1\right)=9^{x}-1$$

Calculate 9 to the power of 8 and get 43046721.

$$43046720\left(43046721+1\right)=9^{x}-1$$

Add 43046721 and 1 to get 43046722.

$$43046720\times 43046722=9^{x}-1$$

Multiply 43046720 and 43046722 to get 1853020188851840.

$$1853020188851840=9^{x}-1$$

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

$$9^{x}-1=1853020188851840$$

Add 1 to both sides of the equation.

$$9^{x}=1853020188851841$$

Take the logarithm of both sides of the equation.

$$\log(9^{x})=\log(1853020188851841)$$

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

$$x\log(9)=\log(1853020188851841)$$

Divide both sides by \log(9).

$$x=\frac{\log(1853020188851841)}{\log(9)}$$

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

$$x=\log_{9}\left(1853020188851841\right)$$

Solve for x (complex solution)

$$x=\frac{\pi n_{1}i}{\ln(3)}+16$$

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