Question

2^(2x-3)=5^(x-2)
Find the exact solution, using common logarithms, and a two-decimal-place approximation of each solution, when appropriate.
Answer: (log(8/25))/(log(4/5))=5.11
We had a sub today and he was terrible, we didn't have an example like this. Can someone help me?? :/ I keep trying it out and I just don't know what I am doing anymore. Please and thank you!

Answer 1

it is a lot of algebra after the first step

Answer 2

first step is
\[(2x-3)\log(2)=(x-2)\log(5)\] then solve for \(x\) with the understanding that \(\log(2)\) and \(\log(5)\) are just some constants

Answer 3

multiply out first, get
\[2\log(2)x-3\log(2)=\log(5)x-2\log(5)\] then put all terms with \(x\) on one side of the equal sign, everything else on the other

Answer 4

\[2\log(2)x-\log(5)x=3\log(2)-2\log(5)\]

Answer 5

factor out the \(x\) from the left hand side, get
\[\left(2\log(2)-\log(5)\right)x=3\log(2)-2\log(5)\] then divide as the last step

Answer 6

\[x=\frac{3\log(2)-2\log(5)}{2\log(2)-\log(5)}\]

Answer 7

that is the answer, the rest is showing off, since \(3\log(2)=\log(2^3)=\log(8)\) and similarly \(2\log(5)=\log(25)\)

Answer 8

so the numerator is \(\log(\frac{8}{25})\) but all that is unnecessary nonsense, it doesn't help compute the number

Answer 9

that also explains why the denominator is \(\log(\frac{4}{5})\)

Answer 10

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