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.

I'm trying to solve the equation $$3^{5x-2}=8^{8x-9}.$$

I'm assuming I need to do some work with logarithms, but I don't know what to do.

Thanks in advance!

share|improve this question
5  
Is this homework? What laws for lograithms do you know? For the formatting: Use curly brackets for exponents ... –  martini Sep 17 '11 at 18:24
2  
Hint: Let us write $\log x$ for the logarithm of $x$ to your favourite base ($10$, $e$, $2$, it doesn't matter). We have in general $\log(a^b)=b\log a$. –  André Nicolas Sep 17 '11 at 18:44
2  
Did you look at the logarithm laws - did you try anything? –  AD. Sep 17 '11 at 18:52
    
this homework isn't going to be marked, it's simply review for my calculus course. I know all of the main log laws (subtracting log, addition of log, changing base, and the a*log(x) = log(x^a) –  KubaSub Sep 17 '11 at 19:11
add comment

3 Answers

\begin{align*} \ln 3^{5x-2} &= \ln 8^{8x-9}\\ \iff \ln 3 \cdot (5x-2) &= \ln 8 \cdot (8x-9) \\ \iff 8\ln 8 \cdot x-5\ln 3 \cdot x &=-2\ln 3+9\ln 8\\ \iff x(8\ln 8-5\ln 3)&=-2\ln 3+9\ln 8\\ \iff x&=\frac{-2\ln 3+9\ln 8}{8\ln 8-5\ln 3}. \end{align*}

share|improve this answer
    
great answer @Andres ,he needs only excel and using function ln he can calculate everything –  dato datuashvili Sep 17 '11 at 18:44
    
thanks a bunch. I was trying this approach, but I used a base of 3 or 8 so I could use one of the other laws and I just ended up confusing myself. This is quite straightforward :) –  KubaSub Sep 17 '11 at 19:16
add comment

Yes, you could apply log's to both sides and then solve the linear equations for $\:x$.

Alternatively, dually, you can trade off knowledge of logs for exponents. Namely rewrite it as

$$ \dfrac{8^{\:9}}{3^{\:2}}\ =\ \bigg(\dfrac{8^{\:8}}{3^{\:5}}\bigg)^x$$

$$ \Rightarrow\quad x\ =\ \dfrac{\log(8^9/3^2)}{\log(8^8/3^5)}\quad\ \ $$

share|improve this answer
add comment

Using base 3 or base 8 will work just as well: $$\begin{align} 3^{5x-2} & =8^{8x-9} \\ \\ 5x-2 & = \log_3 \left(8^{8x-9}\right) = (8x-9)\log_3 8 = 8(\log_3 8)x -9\log_3 8 \\ \\ 5x - 8(\log_3 8)x & = 2 - 9\log_3 8 \\ \\ (5 - 8\log_3 8)x & = 2 - 9\log_3 8 \\ \\ x & = \frac{2 - 9\log_3 8}{5 - 8\log_3 8}. \end{align} $$

share|improve this answer
add comment

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.