# Logarithmic properties

1. Jan 23, 2006

### Rafe

sorry this should go under the " Homework, Coursework, & Textbook Questions" forum. please delete. i will repost.

Okay i did a search for logarithmic properties and logarithms and couldn't seem to find an explanation for how this particular property works.
(log base c of a ) ^ b = b (log base c of a)
when i input simple numbers like :
PHP:
a=4
b=3
c=2
Log base 2 of 4 obvioussly the answer is 2, but
2^3 /= (does not equel) 3 x 2.
i dont' know how to make sense of this discrepency. i imagine i'm just reading it wrong.

Last edited: Jan 23, 2006
2. Jan 23, 2006

### mathman

You must be reading it wrong. Let x=(log base c of a). Then your equation reads x^b=bx, which in general is wrong.

A correct expression is:

(log base c of a^b)=b(log base c of a)

It looks close to what you wrote, so it may be what was meant.

3. Jan 23, 2006

### HallsofIvy

Staff Emeritus
As mathman said, what you have: (log base c of a)^b = b(log base c of a), more simply written as
$$\left(log_c a\right)^b= b log_c a$$
is not true.

Yes, you are reading it wrong. What is true is that
$$log_c\left(a^b\right)= b log_c(a)$$

How you would prove that depends on exactly which definition of $log_c$ you are using.