question about natural logs

[jwpaine]

Hi.

Question1: Why is it when I enter log(27)/log(3) (both in base 10) that my calculator, (when set on exact) reduces it to ln(3)/ln(2)

is the natural log ln a standard when reducing? I find that quite odd... because ln is the inverse of the natural base e.....


Question2: How would I convert log(27)/log(3) into ln(3)/ln(2)

Thanks.

 

[soroban]

Hello, jwpaine!

I don't know exactly how you're entering the problem,
. . but your calculator is wrong.

By the way, how do you know that reduces to \(\displaystyle \frac{\ln3}{\ln2}\)?
. . Is that thing displayed?

Why is it when I enter log(27)/log(3) (both in base 10)
that my calculator (when set on exact), reduces it to ln(3)/ln(2) . Not true!

We have: \(\displaystyle \L\:\frac{\log(27)}{\log(3)} \:=\:\frac{\log(3^3)}{\log(3)} \:=\:\frac{3\cdot\log(3)}{\log(3)} \:=\:3\)

 

[jwpaine]

Hi Soroban!

Thanks for your reply.
I made a typo: I meant to say a different rational... but now I can't even find what one I had

Let me make up a different one, and then restate my question: When my ti-89 reduces logs... if it the answer is irrational (and when the mode is set on exact) it reduces it to ln.
Is reducing an irrational answer to a natural log, ln, a standard for reduction in the mathematical community?

Lets do the sum of two logs:
log(4) + log(3)

My calculator says ln(12)/ln(10)

Question 1: is this a standard for reduction in the mathematical community?
Question 2: How do I preform this odd change of base. Why would I want to go from base 10, to a natural log? It doesn't make sense to me.

how do I go from log(12) to ln(12)/ln(10) and why is this a good/bad/acceptable answer

Thanks a TON!

 

[pka]

This is point Soroban is making: \(\displaystyle \L \log _b (a) = \frac{{\ln (a)}}{{\ln (b)}}\).

Thus \(\displaystyle \L \log (4) + \log (3) = \log (12) = \frac{{\ln (12)}}{{\ln (10)}}\).

This is a soapbox of mine. Many current authors are beginning to use log in any place historically ln has been used. Leonard Gillman (an ex-president of the MAA) introduced me to that possibility in the early 1970’s. Professor Gillman went from the University of Rochester to be chairman at Texas. Now I know that the TI92 is based on Derive as its CAS. There may be some connection there.
BTW: I absolutely agree with doing what Gillman has done. In complex variables there is only one logarithm, log. But here is a little story. About twenty years ago, I was teaching an analysis graduate course for in-service teachers. Among the students was the chairwoman of the chemistry department of a local college. When I gave my little talk why there is only one log, she became so angry that she never came back. It seems that chemist need base 10.

 

[stapel]

I've heard of "log()" meaning "log₁₀()", "log_e()", and "log₂()". In a context such as this forum, it's probably best always to ask and/or to specify.

Eliz.