# Homework Help: Logarithm notation question

1. Aug 20, 2011

### vanmaiden

1. The problem statement, all variables and given/known data
is ln(x)3 the same as saying [ln(x)]3? Also, if there is a difference, which one applies to the exponent being moved into the front of the logarithm as in 3ln(x)?

2. Relevant equations
natural logarithms

3. The attempt at a solution
I just got tripped up on the notation. Normally I see logarithms written as ln(x)3, but when I saw this, I wasn't quite sure if it was the same or not. I wasn't too sure if it was like sin2(x) and [sin(x)]2.

2. Aug 20, 2011

### HallsofIvy

Yes, $ln(x)^3$ means $(ln(x))^3$ as opposed to $ln(x^3)$. (The second form, with parentheses, is preferable as it is clearer.) It is the latter to which we can apply the "law of logarithms": $ln(x^3)= 3ln(x)$.

Look at a numerical example: ln(2)= 0.6931, approximately, so $(ln(2))^3= 0.3330$ while $ln(2^3)= ln(8)= 2.0794= 3ln(2)$.

Last edited by a moderator: Aug 24, 2011
3. Aug 24, 2011

### vanmaiden

Ah, thank you for the great explanation.

4. Aug 24, 2011

### uart

Be aware that you'll also see abbreviated notations without the brackets. Like,

$$\log x^2 = \log(x^2)$$

and

$$\log^2 x = ( \log(x) )^2$$

This type of notation is frequently used with trig functions too.

5. Aug 26, 2011

### vanmaiden

ah thank you for further clarifying!