Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

How can we solve this equation? $$33(1.207)^x = 47(1.547)^x$$

I've done this step so far.$$33x\ln(1.207) = 47x\ln(1.547)$$

This is where I get stuck. Won't moving an $x$ to one side eliminate all the $x$'s?

share|improve this question
4  
The step you did was wrong. From $ab^c=de^c$ you should get $\log a+c\log b=\log d+c\log e$. –  Gerry Myerson Dec 4 '12 at 23:55

3 Answers 3

up vote 4 down vote accepted

When you take the log of an each side of your equation, you need to remember that

"the log of a product is the sum of the logs":

$$\ln \left(a \cdot b^x \right) = \ln a + \ln (b^x) = \ln a + x \ln b$$

So what does that mean for your equation?

$$33(1.207)^x = 47(1.547)^x$$ $$\ln[33(1.207^x)] = \ln[47(1.547^x)]$$ $$ \ln{(33)} + x \ln{(1.207)} = \ln{(47)} + x\ln{(1.547)}$$

Now solve for $x$

Gather constants to the right, multiples of $x$ to the right.

$$x \ln{(1.207)} - x\ln{(1.547)}= \ln{(47)} -\ln{(33)} $$

Factor out $x$:

$$x(\ln(1.207) - \ln{(1.547)}) = \ln{(47)} - \ln{(33)}$$

Now, divide both sides of the equation by $(\ln(1.207) - \ln(1.547))$, and you are left with:

$$ x = \frac{\ln{(47)} -\ln{(33)}}{\ln(1.207) - \ln(1.547)}$$

Then simplify the expression using the property of logs: $$\ln(a) - \ln(b) = \ln\left(\frac ab\right)$$

share|improve this answer
    
I like how you set up this explanation, but I still don't see how to get x by itself. I know you divide, but then you will have (xln(1.207))/(xln(1.547)), right? –  Tyler Zika Dec 5 '12 at 0:09
    
Tyler: Does that help? –  amWhy Dec 5 '12 at 0:17
    
my mind is blown.. yes that helps, thank you! –  Tyler Zika Dec 5 '12 at 0:20

Recall that $$\log \left(a \cdot b^x \right) = \log a + x \log b$$ Hence, in your case, you will get that $$\log (33) + x \log(1.207) = \log (47) + x \log(1.547)$$

share|improve this answer

Method 1: From $33(1.207)^x=47(1.547)^x$, we have $\ln 33+x\ln1.207=\ln 47+x\ln 1.547$, so that $x\ln 1.547-x\ln1.207=\ln 33-\ln 47$. Hence $x=\frac{\ln 33-\ln 47}{\ln 1.547-\ln 1.207}$.

Method 2: From $33(1.207)^x=47(1.547)^x$, we have $\frac{33}{47}=(\frac{1.547}{1.207})^x$, so that $x\ln\frac{1.547}{1.207}=\ln\frac{33}{47}$. Hence $x=\ln\frac{33}{47}/\ln \frac{1.547}{1.207}$.

Of course they can be seen to give the same answer.

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.