How to solve an exponential equation

1. Aug 7, 2014

Mr Davis 97

1. The problem statement, all variables and given/known data

Solve for x: 8100($\frac{7}{6}$)$^{x}$ = 11,000

2. The attempt at a solution

x = log$_{9450}$(11,000) = 1.02

I know this is not the right answer, but I am not sure what I am doing wrong.

2. Aug 7, 2014

vela

Staff Emeritus
Where did the 9450 come from?

EDIT: Never mind. I figured it out.

Just take the natural log of both sides:
$$\ln \left[8100\left(\frac 76\right)^x\right] = \ln 11000.$$ What do you get when you use the properties of logarithms to expand the lefthand side?

(You could divide both sides by 8100 first, and then take the log of both sides.)

3. Aug 7, 2014

Mr Davis 97

Thank you, I solved it. the answer is approximately 2. However, why didn't my previous attempt work? What was wrong about it?

4. Aug 7, 2014

vela

Staff Emeritus
Because $x = \log_{9450} 11000$ corresponds to the exponential equation $9450^x = 11000$, but $9450^x = \left(8100\times \frac 76\right)^x = 8100^x \left(\frac 76\right)^x$. The last expression, though similar, doesn't match what you started with on the lefthand side.

5. Aug 7, 2014

HallsofIvy

It would be simpler, and less error prone, to first divide both sides by 8100:
$$\left(\frac{7}{6}\right)^x= \frac{11000}{8100}= \frac{110}{81}$$

Now take the logarithm of both sides.

6. Aug 9, 2014

HallsofIvy

$$8100\left(\frac{7}{6}\right)^x$$
is NOT the same as
$$\left(8100*\frac{7}{6}\right)^x$$