How to solve an exponential equation

[Mr Davis 97]

Homework Statement

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.

 

[vela]

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.)

 

[Mr Davis 97]

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.)

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

 

[vela]

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.

 

[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.

 

[HallsofIvy]

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