1. ## Natural logarithms

Hi,

I am doing a review on logarithms. When I reached my last question's solution I found something strange.

Question: Solve, Ln(y-1)=1+Ln(3y+2)

The solution gives:
Step 1. Ln(y-1)-Ln(3y+2)=1
Ln((y-1)/(3y+2))=1

Step 2. e^( Ln((y-1)/(3y+2))=e^1

(y-1)/(3y+2)=e

The solution then goes on:

y-1=e(3y+2)=3e*y+2e

(1-3e)y=1+2e

Finally,

y=(1+2e)/(1-3e)=-0.8996

Can you tell me what's wrong here and what does e mean without an index?

Or is there something I have failed to recognise?

2. ## Re: Natural logarithms

Originally Posted by charlesrussell
Hi,

I am doing a review on logarithms. When I reached my last question's solution I found something strange.

Question: Solve, Ln(y-1)=1+Ln(3y+2)

The solution gives:
Step 1. Ln(y-1)-Ln(3y+2)=1
Ln((y-1)/(3y+2))=1

Step 2. e^( Ln((y-1)/(3y+2))=e^1

(y-1)/(3y+2)=e

The solution then goes on:

y-1=e(3y+2)=3e*y+2e

(1-3e)y=1+2e

Finally,

y=(1+2e)/(1+3e)=-0.8996

I found no solution and after graphing saw that there could be no solution for real numbers.

Can you tell me what's wrong here and what does e mean without an index?
How does e become -0.404273?

Or is there something I have failed to recognise?
e is Euler's number, which has the property of being the base of an exponential function so that its derivative is equal to the original function.

3. ## Re: Natural logarithms

You actually have the correct answer, with y = -0.8996 (one minor nit to pick - in the line right after "Finally" you should have y=(1+2e)/(1-3e)=-0.8996).

The issue is that the log of a negative number is a complex number, which has a real part consisting of the log of the absoulte value of that negative number plus an imaginary part equal to pi. This comes from: log(-a) = log(-1 x a) = log (-1) + log (a), and form e^(i pi) =-1 you get log(-1) = i pi. So your solution of -0.8996 when put back into the original equation yields:

ln(y-1) = 1 + ln(3y+2)
ln(-1.8996) = 1 + ln(-0.69883)
ln(1.8996)+i pi = 1 + ln(0.69883) + i pi
0.6416 + i pi = 1 + -0.3584 + i pi
0.6416 + i pi = 0.6416 + i pi

So it checks out.

4. ## Re: Natural logarithms

what does e mean without an index?
Since you are doing a problem involving "ln(x)", I presume you know what that means. It is the inverse function to " $e^x$". And that means just the number "e" to the x power. Saying that is "without an index" means it is $e^1$, just as "x", in a polynomial "without a coefficient" would be "1x".