Subtracting powers with variable in exponent

Question

I am having some troubles with a question that subtracts powers.

Solve for unknown: $$3^{x+4} - 5(3^x) = 684$$

I have a hunch that I should apply factorization somehow. Do I multiply 5 and 3 to begin or should I change to logarithm form and have the 5 act as an exponent on $3^x$, and then factorize?

Any recommendations are appreciated!

Answer 1

I've edited it at the user's notification.

$$3^{x+4}-5(3^x)=684 \implies 3^x\cdot3^4-5(3^x)=684 \implies 3^x(3^4-5)=684 \implies (76)3^x=684 \implies 3^x = \frac{684}{76} \implies 3^x = 9$$

You can take it from here.

Answer 2

I'm not sure if you mean $3^{x+4}-5(3^x)=684$ or $3^x+4-5(3^x)=684$. If you mean the first, then rewrite $3^{x+4}$ as $3^x\cdot3^4$ and then factor $3^x$ out. If you mean the second way, subtract $4$ from each side and then factor out $3^x$. Hope this helps.

Answer 3

You factor out the $3^x:$ $(3^4-5)(3^x)=680$ $\longrightarrow$ $76(3^x)=680$
Now, you can solve this equation with calculator.