Division by log base 10 and ln

**merrysoul** · July 30th 2009, 12:21 AM

Had a question someone might be able to answer...

Why do I get the same number if I divide, let's say log5/log3, as if I divided ln5/ln3???

What exactly is the relationship between logs and natural logs that makes this true?

Thanks in advance.

M

**Gamma** · July 30th 2009, 12:32 AM

Those better not yield the same answer.
log(5)=x where x satisfies $10^x=5$
ln(5)=x where x satisfies $e^x=5$
log(3)=x where x satisfies $10^x=3$
ln(3)=x where x satisfies $e^x=3$

If you got the same answer when you calculated these ratios you have made an error somewhere.

**Gamma** · July 30th 2009, 12:37 AM

If you are interested, you can express logs in different bases in terms of the natural log as follows.

$y=log_a(b)\Rightarrow a^y=b\Rightarrow ln(a^y)=ln(b)\Rightarrow yln(a)=ln(b) \Rightarrow y=\frac{ln(b)}{ln(a)}$

So we see that

$log_a(b)=\frac{ln(b)}{ln(a)}$

**merrysoul** · July 30th 2009, 12:43 AM

Right, that makes sense. I think I understand logs somewhat... The thing is when I was working a problem and I input the data into my calculator, I got the same answer whether I hit the log key or the ln key.

**Gamma** · July 30th 2009, 12:49 AM

hmm now that is strange. generally log without any subscript is understood to be log base 10. Whereas ln is to be log base e. might be worth seeing whether it is giving you ln or log though so you can apply the change of base stuff as necessary. try log(10) and ln(e) if you guess the base right you should get 1. That oughta at least tell you which base it is giving you.

**merrysoul** · July 30th 2009, 12:54 AM

I thought so too, but we did several examples to prove what I don't know, that's what I am trying to figure out...lol.

I have been curious since Monday and I have tried researching all I can about logarithms. I don't have either of my calculators on me but I am going to keep trying to figure it out.

Thanks for your help though. If you have a calculator handy, input the numbers, first using the log key then use the ln key. Weird.

**Gamma** · July 30th 2009, 01:08 AM

oh dude, yeah you are right they should be the same, I was thinking $ln(5/3)$ and $log(5/3)$ my bad dude. Here is why it is true.

$a=log(5)\Rightarrow 10^a=5 \Rightarrow a=\frac{ln(5)}{ln(10)}$
$b=log(3)\Rightarrow 10^b=5 \Rightarrow b=\frac{ln(3)}{ln(10)}$

$\frac{log(5)}{log(3)}=\frac{a}{b}=\frac{\frac{ln(5 )}{ln(10)}}{\frac{ln(3)}{ln(10)}}=\frac{ln(5)}{ln( 3)}$

**stapel** · July 30th 2009, 07:29 AM

Originally Posted by merrysoul

Why do I get the same number if I divide, let's say log5/log3, as if I divided ln5/ln3??

It's the change-of-base formula:

log_b(x) = log_c(x)/log_c(b)

This is because:

log_b(x) = y

x = b^y

log_c(x) = log_c(b^y) = y*log_c(b)

log_c(x)/log_c(b) = y = log_b(x)

By applying the change-of-base formula twice (once backwards and once forwards), we get:

log_10(5)/log_10(3) = log_3(5) = ln(5)/ln(3)

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