How to solve if I have ln on both sides of equation?

Question

I thought this would be a common problem but googling hasn't helped.

If I have $\ln(ex)=\ln(y) $ what the next step to solve for $y$?

It is a very common step to do in mathematics and googling "solve equations with ln" gave me this first link : http://www.sosmath.com/algebra/logs/log4/log47/log47.html — user88595, Apr 29 '14 at 07:02
Yes, but in your particular equation, is the lefthand side $\ln(ex)$ or $\ln(e^x)$? — 5xum, Apr 29 '14 at 07:04
I read the link already, and am familiar with working with logs on one side of the equation, just not both. Under example 3, steps 3,4,5 they show this situation but don't clearly explain the steps, just suddenly the log is gone by "simplifying" — user3550682, Apr 29 '14 at 07:08

score 1 · Accepted Answer · answered Apr 29 '14 at 07:05

1

If you notice that $y = e^{\ln y}$, then you have $y = e^{\ln (ex)}$ as well; but $e^{\ln(ex)} = ex$, so $y = ex$!

And if it is $\ln y = \ln (e^x)$, you can still say $y = e^{\ln y}$ so $y = e^{\ln (e^x)} = e^x$!

Hope this helps! Cheerio,

and as always,

Fiat Lux!!!

answered Apr 29 '14 at 07:05

Robert Lewis

Would there be a caution on the possible range of values if one is working in the real numbers as ln isn't defined for negative values? – JB King Apr 29 '14 at 08:40

score 1 · Answer 2 · answered Apr 29 '14 at 07:06

1

$$\ln a = \ln b\\ e^{\ln a} = e^{\ln b}\\ a = b$$

answered Apr 29 '14 at 07:06

user137794

2 Answers2