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I'm translating this from my mother tongue so bear with me - I have a question in relation to power functions but I think the problem I am dealing with is my seemingly lack of other mathematical rules that I need! The equation I want to solve is:

$$7\cdot 5^x = 9-6\cdot 5^x$$

Now I know I can't just simplify it by subtracting $6$ from $9$, because of the product rule $-6\cdot 5^x$. Then, I tried to put $-6\cdot 5^x$ to the left side:

$$\frac{7\cdot 5^x}{-6\cdot 5^x} = 9$$

I think that's right, since the opposite of a product is division so I don't need to switch up the minus sign. But then I don't know what follows.. could someone please break it down for me

Christian E. Ramirez
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Leatobeya
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  • This is hard to read. Are you asking about $7\times 5^x=9-6\times 5^x$? – lulu Jan 18 '23 at 22:07
  • If so, note that $7\times 5^x+6\times 5^x=13\times 5^x$, from the Distributive Law. – lulu Jan 18 '23 at 22:12
  • yes! not sure if I can directly reply to your comment – Leatobeya Jan 18 '23 at 22:13
  • Your second equation is wrong. It would be correct if the $9$ multiplied the rest of the right member but you can't undo the subtraction by dividing. Hint: divide each of the three terms by $5^x$, or use the hint from @lulu . – Ethan Bolker Jan 18 '23 at 22:14
  • so my first step should be to divide both sides by 5^x? – Leatobeya Jan 18 '23 at 22:22
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    I would add $6 • 5^x$ to both sides, that way you can combine it with $7•5^x$ through the distributive property, then you get an exponential equation which you can solve by taking log base 5 of both sides – Math Man Jan 18 '23 at 22:27

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Let $5^x$ be $y$.

Now the question becomes $$7y = 9-6y$$

This is a linear equation, we would collect all the terms containing $y$ at one side.

$$7y+6y=9$$ You should be able to obtain $$y=\frac{9}{13}.$$

Hence $$5^x=\frac9{13}$$

Now use logarithm to solve your question.

Siong Thye Goh
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  • that was useful, but wouldn't I have 5^x ⋅ 5^x left? because 7⋅5^x+6⋅5^x =9, then I add 7+6 and put it on the side ie 5^x ⋅ 5^x = 9/13 – Leatobeya Jan 20 '23 at 10:44
  • Nope, we wouldn't have $5^x$ and $5^x$ multiplied together. If we sum $7$ apples and $6$ apples, we get $13$ apples. – Siong Thye Goh Jan 20 '23 at 12:48