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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!

jdods
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randyslavage
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  • You can't multiply the 5 and 3 because of pemdas. Exponentiation to x comes before multiplication. To proceed, realize that 3^x and 5(3^x) are like terms. Combine them. – ziggurism May 20 '16 at 03:37

3 Answers3

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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.

B.H.
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  • After I submitted my comment I now see the equation formatted correctly, so I see that you actually meant the second way. – B.H. May 20 '16 at 04:16
  • You get a suspiciously nice value of $x$ when you use the first interpretation , so I would go with that over an imaginary solution, which is what you get if you go with the second one. – WW1 May 20 '16 at 04:18
  • indeed. How often have many of us ended up with a more difficult problem because we simply read and copied the problem incorrectly? – B.H. May 20 '16 at 04:30
  • @B.H. you are right, it is the first interpretation! I should have used brackets when formatting!! – randyslavage May 20 '16 at 16:34
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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.

Drew Christensen
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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.

Suganth Kannan
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