Question 363175
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I have to assume that you meant:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ 3^{3x\,-\,4}\ =\ 5^x]


Since the other (equally valid per your notation) interpretation, namely:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ 3^{3x}\ -\ 4\ =\ 5^x]


Can only be solved by numerical methods and therefore an exact solution cannot be determined.


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ 3^{3x\,-\,4}\ =\ 5^x]


Step 1:  Take the log of both sides.  Natural log or base 10 (or any other base you like), it doesn't matter.


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ \ln\left(3^{3x\,-\,4}\right)\ =\ \ln\left(5^x\right)]


Use


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ \log_b(x^n)\ =\ n\log_b(x)]


to write:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ (3x\ -\ 4)\ln\left(3\right)\ =\ x\ln\left(5\right)]


Multiply both sides by *[tex \Large \frac{1}{ln(3)}]


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ 3x\ -\ 4\ =\ \frac{x\ln\left(5\right)}{\ln\left(3\right)}]


Add *[tex \Large 4] and  *[tex \Large -\frac{x\ln\left(5\right)}{\ln\left(3\right)}] to both sides:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ 3x\ -\ \frac{x\ln\left(5\right)}{\ln\left(3\right)}\ =\ 4]


Factor out an *[tex \Large x]


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ x\left(3\ -\ \frac{\ln\left(5\right)}{\ln\left(3\right)}\right)\ =\ 4]


Multiply both sides by *[tex \Large \frac{1}{3\ -\ \frac{\ln\left(5\right)}{\ln\left(3\right)}}]


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ x\ =\ \frac{4}{3\ -\ \frac{\ln\left(5\right)}{\ln\left(3\right)}}]




John
*[tex \LARGE e^{i\pi} + 1 = 0]
My calculator said it, I believe it, that settles it
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