Simple logarithm question: simplify log(10)^5/log(10)^2 as log(10)^3

[evking]

Why is it wrong, for example, to simplify log(10)⁵/log(10)² as log(10)³

 

[Steven G]

Why is it wrong, for example, to simplify log(10)⁵/log(10)² as log(10)³

If you had (log(10))⁵/(log(10))², then it would simplify to (log(10))³ .For the record, my equation reduces to 1⁵/1² =1³=1

What you have log(10)⁵/log(10)² = [5log(10)]/[2log(10)] = 5/2

 

[HallsofIvy]

Why is it wrong, for example, to simplify log(10)⁵/log(10)² as log(10)³

It's not wrong. For any non-zero number, a, \(\displaystyle \frac{a^5}{a^2}= a^3\). Of course, if the logarithm here is base 10, then log(10)= 1 so this would be more simply written as "1"!

 

[Steven G]

It's not wrong. For any non-zero number, a, \(\displaystyle \frac{a^5}{a^2}= a^3\). Of course, if the logarithm here is base 10, then log(10)= 1 so this would be more simply written as "1"!

Be careful, the powers are NOT for log10, but rather for just the 10. The answer should be 5/2.

 

[Steven G]

WHY do you say it's "wrong"?
Havva look:
http://www.wolframalpha.com/input/?i=log(10)^5/log(10)^2=log(10)^x

Because it is wrong. I completely disagree with wolfamalpha as they rewrote the problem incorrectly. I maintain that those powers are for the 10s and not for log10.



Please my post above and tell me where my error is.

 

[Dr.Peterson]

Because it is wrong. I completely disagree with wolfamalpha as they rewrote the problem incorrectly. I maintain that those powers are for the 10s and not for log10.

The notation is somewhat ambiguous, but I would say that their interpretation is at least as valid as yours, and certainly consistent with how a programmer would look at it.

If you want to express the log of a power, you can write either \(\displaystyle \log x^n\) or \(\displaystyle \log (x^n)\) . If you want to express a power of a log, you should write \(\displaystyle (\log x)^n\) or, as they do but I would avoid, \(\displaystyle \log^n x\).

When you write \(\displaystyle \log (x)^n\), I would take "log(x)" as a unit, because the parentheses naturally indicate the argument, and then take the power of that; but I would ask if that is what is intended, because it is easy to see it as (x) raised to a power, and the whole used as the argument.

The ambiguity is due to the old tradition of not using parentheses around the arguments of special functions like log and sin. If we consistently used parentheses, as programmers do, then there would be no question.

The real question we need to ask is, what did the OP mean by it? All that is wrong, in my mind, is to have written it that way in the first place.

 

[evking]

The notation is somewhat ambiguous, but I would say that their interpretation is at least as valid as yours, and certainly consistent with how a programmer would look at it.

If you want to express the log of a power, you can write either \(\displaystyle \log x^n\) or \(\displaystyle \log (x^n)\) . If you want to express a power of a log, you should write \(\displaystyle (\log x)^n\) or, as they do but I would avoid, \(\displaystyle \log^n x\).

When you write \(\displaystyle \log (x)^n\), I would take "log(x)" as a unit, because the parentheses naturally indicate the argument, and then take the power of that; but I would ask if that is what is intended, because it is easy to see it as (x) raised to a power, and the whole used as the argument.

The ambiguity is due to the old tradition of not using parentheses around the arguments of special functions like log and sin. If we consistently used parentheses, as programmers do, then there would be no question.

The real question we need to ask is, what did the OP mean by it? All that is wrong, in my mind, is to have written it that way in the first place.

Thanks for clearing things up. The syntax is what confused me, as you have mentioned, since I wasn't sure if the question was expressing the log of a power or a power of a log.