Expert: Paul Klarreich - 3/14/2008

Question
hi Paul, i have been asked to solve 2^(33-x)=3^(2x-1) for x. i have taken logs to both sides and rearranged to get (33-x)/(2x-1)=log3/log2 but cant seem to get the right x value to 5 decimal places, i know its between 8.2 and 8.3. just wondering where to go next. your help would be much appreciated. thank you. Adam

Answer
Questioner:   adam
Category:  Advanced Math
Private:  No
 
Subject:  logs
Question:  hi Paul, i have been asked to solve 2^(33-x)=3^(2x-1) for x. i have taken logs to both sides and rearranged to get (33-x)/(2x-1)=log3/log2 but cant seem to get the right x value to 5 decimal places, i know its between 8.2 and 8.3. just wondering where to go next. your help would be much appreciated. thank you. Adam
............................  
Hi, Adam,

While I work on this, there is something else I can help you with -- your computer keyboard.  There are these large keys labeled 'Shift'. If you hold one of them down while you type 'i', you get 'I'.  If you do that, your text does not look as if it was typed by a teenager.  This will be very useful when you write college essays or resumes.

Now, about 2^(33-x)=3^(2x-1)

2^(33-x) = e^(33 - x)ln 2, and

3^(2x - 1) = e^(2x - 1)ln 3, so

(33 - x)ln 2 = (2x - 1)ln 3

[So far, so good.]

33 ln 2 - x ln 2 = 2 x ln 3 - ln 3  << multiply out.

33 ln 2 - ln 3 = 2x ln 3 + x ln 2  << rearrange terms

33 ln 2 - ln 3 = x(2 ln 3 +  ln 2) << factor out x.

x = (33 ln 2 - ln 3)/(2 ln 3 +  ln 2) << divide by coeff. of x.

Looks like just some calculator stuff o me.  Maybe you just plug this into Excel -- that might be easiest.

Type it as:
=(33*LN(2) - LN(3))/(2*LN(3)+LN(2))
which comes to 7.53371763 on my machine.