How can different types of logarithms affect the derivative of a function?

[nofunatall]

How do you find the derivative of the following?

e^5X - 3log(x)

 

[Zurtex]

Do you know how to use the chan rule and what the derivative of a log is?

 

[nofunatall]

Do you know how to use the chan rule and what the derivative of a log is?

Learned years ago. In other words. No.

 

[Zurtex]

Well:

$$\frac{d}{dx} ( \log x ) = \frac{1}{x} \quad \text{for all} \, x > 0$$

The chain rule goes:

$$\frac{dg}{dx} = \frac{dg}{df} \frac{df}{dx}$$

Or using Newton notation, if there exists some y = g(f(x)) then y' = f'(x) g'(f(x))

 

[nofunatall]

Got that part. That makes sense.

 

[bomba923]

Well:
$$\frac{d}{dx} ( \log x ) = \frac{1}{x} \quad \text{for all} \, x > 0$$

~No, that's not true;

$$\forall x > 0, \; \frac{d}{{dx}}\ln x = \frac{1}{x}$$

$$\forall x > 0, \; \frac{d}{{dx}}\log x = \frac{1}{{x\ln 10}}$$

The natural logarithm of $x$ is written as $\ln(x)$, not $\log(x)$.

log(x) is the base 10 logarithm,
$$\log x = \frac{{\ln x}}{{\ln 10}}$$

 

[Moo Of Doom]

The natural logarithm of $x$ is written as $\ln(x)$, not $\log(x)$.

Not in all cases. For example, it is usual in analysis to use just log to mean base e. Sadly, this does cause some confusion, so people really should write the base when there's no context.

 

[Zurtex]

Hmm, I've not seen log mean log₁₀ in a good year or so now, I'm so used to log meaning log_e I just assumed this was the case. I still think it does, but only the original poster will be able to tell us.

 

[matt grime]

log should always be taken to mean log base e unless in certain very strict cases none of which are applicable in anaysis. certainly very few people in mathematics would ever write ln for natural log unless close by they had a need to use logs in other bases (see below)short of the occasional use in engineering/applied maths no one uses base 10, and in fact the most natural second choice after e ought to be base 2.

 

[0rthodontist]

My algorithms textbook uses three different kinds of logarithm, log, ln, and lg for base 2. I thought that was especially unusual because in algorithms it usually doesn't matter what base you're using.

 

[arildno]

My algorithms textbook uses three different kinds of logarithm, log, ln, and lg for base 2.

How horrid, and utterly dumb.

I still remember how shocked I was in a class of fluid mechanics where my professor almost apologetically said that a particular formula used Briggsian logarithms rather than the natural one.
(It was a typical "engineer" formula).