Finding x in a Logarithmic Equation: A Challenging Problem

[nvez]

Homework Statement

This one has has me pretty lost, I'm not at all sure what to do, I've tried a couple of things. I have to find x.

2^(x-1) = 5(7^3x)

Homework Equations

Logarithm laws?

The Attempt at a Solution

(did multiple attempts but my most reasonable to my account)

2^(x-1) = 5(7^3x)
[2^(x-1)/5] = 7^3x
[(x-1) log 2/5] = 3x log 7
(x-1) log 2 = 5(3x*log 7)
(x-1) log 2 = 15x * 5 log 7
(x-1) log 2 = 15x * log 7⁵
(x-1) log 2 = 15x * log 16807
x-1 = (15x * log 16807) / (log 2)
x-1 = 15x * 14.034
x = 15x * 14.034 + 1
x = 15x * 15.034
x/15x = 15.034
1/15x = 15.034
x = 15.034 * (1/15)
x = 210.51

I believe I am completetly off and I really can't figure this out, help will be appreciated!

 

[BobMonahon]

Hi,
Try taking the log right away:

(x-1)ln2 = ln5 + 3xln7
xln2 - 3xln7 = ln5 + ln2
x = (ln5 + ln2) / (ln2 - 3ln7)

Not a very satisfying answer, is it?

Regards, BobM

 

[nvez]

Hi,
Try taking the log right away:

(x-1)ln2 = ln5 + 3xln7
xln2 - 3xln7 = ln5 + ln2
x = (ln5 + ln2) / (ln2 - 3ln7)

Not a very satisfying answer, is it?

Regards, BobM

Thank you, that is very helpful, I understand how you got from
xln2 - 3xln7 = ln5 + ln2
to
x = (ln5 + ln2) / (ln2 - 3ln7)

but I don't really understand what happened with the (x-1) when you go from
(x-1)ln2 = ln5 + 3xln7
to
xln2 - 3xln7 = ln5 + ln2

Thank you in advanced.

 

[nvez]

I understood what was done!

(x-1)ln2 = ln5 + 3xln7
x ln 2 - ln 2 = ln 5 + 3xln7
x ln 2 - 3x ln 7 = ln 5 + ln 2
x (ln 2 - 3 ln 7) = ln 5 + ln 2
x = ln 5 + ln 2 / ln 2 - 3 ln 7

Thanksss!