How to differentiate $\mathrm{ln}({a}^{x})$?

Can someone give me the process to differentiate this (with respect to x)?

$$\mathrm{ln}({a}^{x})$$

Can someone give me the process to differentiate this (with respect to x)?

$$\mathrm{ln}({a}^{x})$$

Christopher Saunders
2022-10-17
Answered

How to differentiate $\mathrm{ln}({a}^{x})$?

Can someone give me the process to differentiate this (with respect to x)?

$$\mathrm{ln}({a}^{x})$$

Can someone give me the process to differentiate this (with respect to x)?

$$\mathrm{ln}({a}^{x})$$

You can still ask an expert for help

Marlene Welch

Answered 2022-10-18
Author has **23** answers

Since

$$\mathrm{ln}({a}^{x})=x\mathrm{ln}a,$$

you can see that this function is in fact linear in $x$. (Note that $\mathrm{ln}a$ is just some constant.) The derivative is therefore $1\cdot \mathrm{ln}a=\mathrm{ln}a.$

$$\mathrm{ln}({a}^{x})=x\mathrm{ln}a,$$

you can see that this function is in fact linear in $x$. (Note that $\mathrm{ln}a$ is just some constant.) The derivative is therefore $1\cdot \mathrm{ln}a=\mathrm{ln}a.$

fluerkg

Answered 2022-10-19
Author has **3** answers

Hint:

Use the fact that

$${a}^{x}={e}^{x\mathrm{ln}a}.$$

Use the fact that

$${a}^{x}={e}^{x\mathrm{ln}a}.$$

asked 2022-08-22

Trouble evaluating the sum involving logarithm

I was trying to solve this problem: Closed form for ${\int}_{0}^{1}\mathrm{log}\mathrm{log}(\frac{1}{x}+\sqrt{\frac{1}{{x}^{2}}-1})\mathrm{d}x$

In the procedure I followed, I came across the following sum:

$\sum _{k=1}^{\mathrm{\infty}}(-1{)}^{k-1}k(\frac{\mathrm{ln}(2k+1)}{2k+1}-\frac{\mathrm{ln}(2k-1)}{2k-1})$

I cannot think of any approaches which would help me in evaluating the sum.

Any help is appreciated. Thanks!

I was trying to solve this problem: Closed form for ${\int}_{0}^{1}\mathrm{log}\mathrm{log}(\frac{1}{x}+\sqrt{\frac{1}{{x}^{2}}-1})\mathrm{d}x$

In the procedure I followed, I came across the following sum:

$\sum _{k=1}^{\mathrm{\infty}}(-1{)}^{k-1}k(\frac{\mathrm{ln}(2k+1)}{2k+1}-\frac{\mathrm{ln}(2k-1)}{2k-1})$

I cannot think of any approaches which would help me in evaluating the sum.

Any help is appreciated. Thanks!

asked 2022-07-20

Proof ' that $\mathrm{ln}(x)$ converges

Where is the flaw in the following 'proof '?

$\underset{x\to \mathrm{\infty}}{lim}\left[\frac{\mathrm{d}}{\mathrm{d}x}\{\mathrm{ln}(x)\}\right]=\underset{x\to \mathrm{\infty}}{lim}\left[\frac{1}{x}\right]=0\phantom{\rule{thickmathspace}{0ex}}\u27f9\phantom{\rule{thickmathspace}{0ex}}\underset{x\to \mathrm{\infty}}{lim}[\mathrm{ln}(x)]=\text{constant}\in \mathbb{R},$

therefore $\mathrm{ln}(x)$ converges to some real number. $\u25fb$

Where is the flaw in the following 'proof '?

$\underset{x\to \mathrm{\infty}}{lim}\left[\frac{\mathrm{d}}{\mathrm{d}x}\{\mathrm{ln}(x)\}\right]=\underset{x\to \mathrm{\infty}}{lim}\left[\frac{1}{x}\right]=0\phantom{\rule{thickmathspace}{0ex}}\u27f9\phantom{\rule{thickmathspace}{0ex}}\underset{x\to \mathrm{\infty}}{lim}[\mathrm{ln}(x)]=\text{constant}\in \mathbb{R},$

therefore $\mathrm{ln}(x)$ converges to some real number. $\u25fb$

asked 2022-07-23

Solve the equation for exact answer. ${9}^{3x+5}=18$

asked 2022-11-06

Expanding logarithm of function

Is there a way (there has to be), I can expand an expression like this?

$${\mathrm{log}}_{2}(3f(n{)}^{n})$$

P.S. This part of an assignment I'm working on, please do not give solutions

Is there a way (there has to be), I can expand an expression like this?

$${\mathrm{log}}_{2}(3f(n{)}^{n})$$

P.S. This part of an assignment I'm working on, please do not give solutions

asked 2022-01-20

Closed form for the partial sum $\sum _{k=1}^{n}\frac{\mathrm{ln}k}{k}$

asked 2021-10-25

Change the logarithmic statement to an equivalent statement involving an exponent.

$\mathrm{ln}x=4$

asked 2022-09-04

If $f(x)={x}^{x}\mathrm{ln}(5x-5)$ , and ${f}^{\prime}(x)={x}^{x}\mathrm{ln}(5x-5)(g(x))$, Then What Would g(x) Equal?