Logs and inequalities

• July 28th 2011, 04:34 PM
JamesBond16
Logs and inequalities
My study guide states that log_2x-log_3(4x-5)>1 is x>67.7721 but I have yet to rap my head around logs. If someone could reveal to me the process behind this it would greatly appreciated.
• July 28th 2011, 04:34 PM
Prove It
Re: Logs and inequalities
Start by converting them to the same base...
• July 28th 2011, 04:40 PM
JamesBond16
Re: Logs and inequalities
Would be to much trouble to ask you to show me?

is it okay to leave it as a inequality?

Since the base is the same, you can actually make this log (2x/(3(4x-5))) >1 or
log (2x/(12x-15)) >1. And this simply means that (2x/12x-15)>10.

Temporarily assuming that 12x-15 is positive (x>1.25), you can multiply each side by this and get
2x>120x -150. Then 2x +150>120x, then 150>118 x. Then x<150/118 (but still greater than 1.25).

Temporarily assuming that 12x-15 is negative (x<1.25), you can do the same stuff with the sign pointing the other way, 150<118x , then x>150/118, but this is a contradiction.

So I'm sticking with 1.25<x<150/118 (this fraction is approx 1.271 btw, so a narrow value range)

http://www2.wolframalpha.com/Calcula...=23&w=162&h=40
• July 28th 2011, 08:10 PM
SammyS
Re: Logs and inequalities
Yes, $\frac{5}{4} is the correct solution for $\log_{10}(2x)-\log_{10}(3(4x-5))>1$

I sure that Prove It assumed that what you meant by log_2x was $\log_2(x)$, etc. The underscore symbol, "_" , generally means that what follows is a subscript.
• July 28th 2011, 08:31 PM
JamesBond16
Re: Logs and inequalities
how how do i get x>67.7721 from this?
• July 28th 2011, 09:34 PM
SammyS
Re: Logs and inequalities
Quote:

Originally Posted by JamesBond16
how how do i get x>67.7721 from this?

I tried to solve it assuming that the bases were 2 & 3 as Prove It reasonably assumed. This does give the approximate answer: x>67.7721.

Assume: $\log_{2}(x)-\log_{3}(4x-5)>1\,.$

Changing the bases gives: $\frac{\ln(x)}{\ln(2)}-\frac{\ln(4x-5)}{\ln(3)}>1\,.$

Multiply both sides by ln(2) and ln(3).

Use each side as the exponent with base e.

Solve it graphically.