Thread: Solving equation with negative exponential with logarithm

1. 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.

2. 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*}

3. 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:
$\displaystyle \ln(x^y) = y \ln x$

4. 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.

5. 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?

6. 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.

7. Re: Solving equation with negative exponential with logarithm

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