Question 609726: 3^n=[square root of]5^n-2. Solve for n.
Answer by solver91311(24713)   (Show Source):
You can put this solution on YOUR website!
})
Square both sides:
})
Take the log of both sides (any base, doesn't matter. If you have a calculator that will calculate logs to any base, or if you can use Excel, you will simplify things greatly if you choose either 3 or 5 as your log base, otherwise you can just use either natural logs or base 10. I'll proceed using natural logs.)
\ =\ \ln\left(5^{(n\,-\,2)}\right))
Use
\ =\ n\log_b(x))
\ =\ (n\ -\ 2)\ln\left(5\right))
A little algebra music, Sammy
\ -\ \ln(5)\right)\ =\ -2\ln(5))
Use
backwards...
\ -\ \ln(5)\right)\ =\ -2\ln(5))
The difference of the logs is the log of the quotient
\right)\ =\ -2\ln(5))
And finally,
}{\left(\ln\left(\frac{9}{5}\right)\right)})
You can do the calculator work from here, or just leave it like that for an exact answer. If you are going to leave it in terms of logs, then go ahead and change the base to 5, using \ =\ 1)
\right)})
John
My calculator said it, I believe it, that settles it
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