# What is the derivative of $\ln(4^x)$?

What is the derivative of $\ln(4^x)$ (which I believe is also equal to $x\ln4$)?

Is it $\dfrac{1}{x\ln4}$?

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as you said, $ln(4^x)$ = $xln4$, so its derivative is simply $ln4$. – the L May 15 '13 at 7:04
or did you mean $\left(\ln{4} \right)^x$ ? – DanZimm May 15 '13 at 7:05
See this page for an explanation of how to write math here. – Zev Chonoles May 15 '13 at 7:05
Sorry, I have now edited the post to be more clear. – Gannicus May 15 '13 at 7:08
@Inceptio Thanks for the dfrac, I didn't know about that LOL – Gannicus May 15 '13 at 7:16

Recall that for $a>0$, we have $$\log(a^b) = b \log(a)$$ Also, note that $$\dfrac{d}{dx}\left( cx\right) = c$$ I trust you can finish it from here.

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The concept of ln, e, log always boggles me. I keep forgetting that they are constants and not scary math figures. – Gannicus May 15 '13 at 7:19

if your question is $ln(4^x)$ then its answer will be $\ ln4$ and if your question is $(\ln4)^x$ then use this formula $$\dfrac {d}{dx}a^x=a^x\cdot\ln{a}$$ so $$\dfrac {d}{dx}{(\ln4)}^x=(\ln4)^x\cdot\ln{(\ln4)}$$

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I've given answer of both possibilities of question so please check carefully who downvote it – iostream007 May 15 '13 at 7:19
(+1). For providing both instances. The down-voter should definitely consider leaving a comment. – Inceptio May 15 '13 at 7:24
peoples are too fast to downvote without reading whole thing and don't want to write reason – iostream007 May 15 '13 at 7:26

Hint : $\ln 4^x=y \implies x \ln 4=y$

Since $\ln4$ is a constant. The derivative is simply the constant, i.e $\ln 4$

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The thing you are supposed to understand here is that constants don't do anything to derivatives -- they just factor out of the expression. More precisely, we have for any function $f$ and a constant $c$ not depending on $x$ that $$\frac{d}{d x} c f(x) = c \frac{d}{d x} f(x)$$ For example, since $\frac{d}{d x} (x^4) = 4x^3$, I can multiply by any constant and the derivative is just as straightforward: $$\frac{d}{d x} \left[ 2.33\pi^2 \cos (4) \right] x^4 = \left[ 2.33\pi^2 \cos (4) \right] 4x^3$$

This works since the $2.33\pi^2 \cos (4)$ is constant - it doesn't depend on $x$.

If you get this, you should be able to calculate derivatives like this easily.

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