Dumb-ass algebra question

Factual Questions
1

Ok… I’m having an algebretarded moment…

If I have 1.1^x / 1.2^x = 5/7, how do I solve for x?

(and no, I’m not a high school student doing his homework! I’m a grad student working on his homework who can’t remember his algebra from 13 years ago)

2

OK, I’m gonna feel really stupid if I’m wrong but I dont’ think that’s algebra. I think yer gona hafta use differential equations to solve that. (and I dont’ remember any of that without a textbook handy but I’m sure a more mathematically gifted Doper wil be along shortly to either tell me I’m wrong or enlighten us…)

3

Could I be a bigger moron? (completely ignore that first post-ARGH! I want edit!) After thinking about it for half a second:

1.1^x/1.2^x = 5/7 is the same as 1.1^x = 5 and 1.2^x = 7. So break up the equations and solve for x. Should be the exact same number for both equations.

4

Or you could write 1.1[sup]x[/sup] / 1.2[sup]x[/sup] = (1.1/1.2)[sup]x[/sup] and do exciting things with logarithms to solve for x.

5

You can’t break it into two equation like that.

1.1^x/1.2^x = (1.1/1.2)^x = (11/12)^x

Now, as g8rguy says, to solve (11/12)^x = 5/7, you need to use logarithms. If you don’t remember how to work with these, you’ll have to remind yourself.

log(11/12)^x = log(5/7)
x log(11/12) = log(5/7)
x = log(5/7) / log(11/12) = 3.8670

If you’re more comfortable saying, instead, x = log[sub]11/12/sub, that’s the same thing. I prefer the first way, though.

6

I plugged the equation into MathCad, out of curiosity, and got the same result as Achernar. It came out to 3.8669912862235766523 (is it actually rational, or did MathCad round it off?)

7

(1.1)^x = 5
(1.2)^x … 7

(1.1/1.2)^x = (5/7)

ln (1.1/1.2)^x = ln (5/7)

x ln (1.1/1.2) = ln (5/7)

x = ln (1.1/1.2)/ln(5/7)

8

OK, I just spent two minutes working through it myself and proving conclusively that I’m a total dumbass. :o Yes, I was TOTALLY wrong and henceforth swear to never answer a math question I haven’t really looked at. (ARGH, my cheeks are literally burning. I shall slink away in shame…)

9

It’s not rational. I rounded too.

10

Argh.

That final step should be: x = ln(5/7)/ln (1.1/1.2)

Looks like others beat me to it anyway.

You can use Base 10 Logs (log) or Base e Natural Logs (ln). You’ll get the same answer.

11

Thanks guys!

I thought it might be something like that, but I don’t have an algebra book anymore, and don’t even know anyone who does.

12

It’s transcendental.