Solving equation with negative exponential with logarithm

Hi guys!

I'm struggling with this problem, I'm sure that the solution is not too difficult, but I just can't figure it out.

6^{-2x} – 15 = 0. Now it is pretty easy to solve that equation the "regular" way and come to the answer of 540, but how do I solve it by using logarithm? If the exponential was positive that would also be easy, but it being negative is really screwing with me. Thanks for the help.

Re: Solving equation with negative exponential with logarithm

I'm pretty sure that $\displaystyle \displaystyle \begin{align*} 6^{-2 \cdot 540} - 15 \end{align*}$ does NOT equal 0...

$\displaystyle \displaystyle \begin{align*} 6^{-2x} - 15 &= 0 \\ 6^{-2x} &= 15 \\ -2x &= \log_6{(15)} \\ x &= -\frac{1}{2}\log_6{(15)} \end{align*}$

Re: Solving equation with negative exponential with logarithm

Hi ManIsuck! :)

$\displaystyle 6^{-2x} - 15 = 0$

$\displaystyle 6^{-2x} = 15$

$\displaystyle \ln (6^{-2x}) = \ln 15$

$\displaystyle (-2x) \ln 6 = \ln 15$

Can you take it from there?

As you can see, the logarithm is never taken from a negative number.

Btw, do you know which rules apply to calculating with logarithms?

Because if you don't that makes it that much harder.

In this particular calculation I used that:

$\displaystyle \ln(x^y) = y \ln x$

1 Attachment(s)

Re: Solving equation with negative exponential with logarithm

I must be missing something. If the "answer" is the value of x that makes the equation true, then I can see at a glance that it must be a number between -1 and 0. How do you get 540? Can you tell me what is the "regular" way? Using natural logarithms, I find x is approximately -0.75570. See the attachment.

Re: Solving equation with negative exponential with logarithm

Thanks for the help guys. I'm not sure where the difference is coming from. I ran the equation through wolfram to check the answer. Maybe I'm using some sign in a different way than you guys?

http://i45.tinypic.com/ulvr8.jpg

Re: Solving equation with negative exponential with logarithm

Can you add a couple of parentheses around (-2x)?

Afterward, check how your "input" looks and whether it matches your problem statement.

Re: Solving equation with negative exponential with logarithm

Thanks. Certainly seems like I picked the right nick. :/