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!