Solve $2^{x+5} - 2^{x+2} = 7$

Question

I have a logarithms related problem at hand which I need to know its related formulas.

$$2^{x+5} - 2^{x+2} = 7$$

I already know the answer which is: $-2$.

So far, I have reached this form:

$$x = \frac{\ln(7 + 2^x+2)}{\ln (2)} - 5.$$

Any tips or hints will be appreciated. Thanks in advance.

Answer 1

Note that $2^{x+5} = 2^x\cdot 2^5 = 32\cdot 2^x$ and $2^{x+2} = 2^x\cdot 2^2 = 4\cdot 2^x$ so your expression becomes

$$ 32\cdot 2^x - 4\cdot 2^x = 7.$$

Can you take it from here?

Answer 2

Hint: write it as $\;2^{x+2}\,(2^3-1)=7\,$.