Question

Use the equation, 8^2x = 32^x+3 , to complete the following problems.

(a) Rewrite the equation using the same base.
(b) Solve for x. Write your answer in the simplest form.

Side note: For your answers, I ask that you show your work so that I can review it and hopefully understand how to do this myself in the future!

Answer 1

Question (a)

Given equation:

`8^(2x) = 32^(x+3)`

8 can be written as
`2^3`

32 can be written as
`2^5`

Therefore, we can rewrite the equation with base 2:

`\implies (2^3)^(2x) = (2^5)^(x+3)`

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Question (b)

To solve:

`(2^3)^(2x) = (2^5)^(x+3)`

Apply the exponent rule
`(a^b)^c=a^(bc)` :

`\implies 2^(3 \cdot 2x) = 2^(5(x+3))`

`\implies 2^(6x) = 2^(5x+15)`

`\textsf{If }a^(f(x))=a^(g(x)), \textsf{ then } f(x)=g(x)` :

`\implies 6x = 5x+15`

Subtract
`5x` from both sides:

`\implies x = 15`