SOLUTION: Use algebraic procedures to find the exact solution(s) of the equation. (Enter your answers as a comma-separated list. If there is no solution, enter NO SOLUTION.)
4^(4x −
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Question 628010: Use algebraic procedures to find the exact solution(s) of the equation. (Enter your answers as a comma-separated list. If there is no solution, enter NO SOLUTION.)
4^(4x − 3) = 5^(−x − 1)
x=(?,?)
Answer by solver91311(24713) (Show Source): You can put this solution on YOUR website!
}\ =\ 5^{(-x\ -\ 1)})
Take the log of both sides (any base, doesn't matter)
}\right)\ =\ \ln\left(5^{(-x\ -\ 1)}\right))
Use \ =\ n\log_b(x))
on both sides
\ln\left(4\right)\ =\ (-x\ -\ 1)\ln\left(5\right))
Multiply both sides by })
\ =\ (-x\ -\ 1)\frac{\ln\left(5\right)}{\ln\left(4\right)})
just for the sake of neatness, let }{\ln\left(4\right)})
\ =\ (-x\ -\ 1)k)
Then it is just plain old algebra from here:
}{\ln\left(4\right)}}{4\ +\ \frac{\ln\left(5\right)}{\ln\left(4\right)}})
Since you want the exact answer, and you don't care about actually being able to approximate the log values, you can get a much neater answer using base 4 logs:
}{4\ +\ \log_4(5)})
I'll let you figure out why these two answers are the same. Hint: What is the value of )
?
John
My calculator said it, I believe it, that settles it
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