Solve 2^x-(x-7)=(2^3)^y+1

$$2 ^ { x - ( x - 7 ) } = ( 2 ^ { 3 } ) ^ { y } + 1$$

Solve for y

$$y = \frac{\log_{2} {(127)}}{3} = 2.3295615622573886$$

Steps Using Definition of Logarithm

To find the opposite of x-7, find the opposite of each term.

$$2^{x-x+7}=\left(2^{3}\right)^{y}+1$$

Combine x and -x to get 0.

$$2^{7}=\left(2^{3}\right)^{y}+1$$

Calculate 2 to the power of 7 and get 128.

$$128=\left(2^{3}\right)^{y}+1$$

Calculate 2 to the power of 3 and get 8.

$$128=8^{y}+1$$

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

$$8^{y}+1=128$$

Subtract 1 from both sides of the equation.

$$8^{y}=127$$

Take the logarithm of both sides of the equation.

$$\log(8^{y})=\log(127)$$

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

$$y\log(8)=\log(127)$$

Divide both sides by \log(8).

$$y=\frac{\log(127)}{\log(8)}$$

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

$$y=\log_{8}\left(127\right)$$

Solve for x

$$x\in \mathrm{R}$$

$$y = \frac{\log_{2} {(127)}}{3} = 2.3295615622573886$$