Is ln(x)3 the Same as [ln(x)]3? Decoding Logarithm Notation

[vanmaiden]

Homework Statement

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)?

Homework Equations

natural logarithms

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.

 

[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 $ln(2^3)= ln(8)= 2.0794= 3ln(2)$.

 

[vanmaiden]

Yes, $ln(x)^3$ means $(ln(x))^2$ 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 $ln(2^3)= ln(8)= 2.0794= 3ln(2)$.

Ah, thank you for the great explanation.

 

[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.

 

[vanmaiden]

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.

ah thank you for further clarifying!