Recent questions in Logarithms

Logarithms
Answered

Conrad Beltran
2022-09-27

Using the method shown here, I have found the following closed form.

${\int}_{0}^{\phantom{\rule{negativethinmathspace}{0ex}}\frac{\pi}{2}}\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}{(\frac{1}{\mathrm{log}(\mathrm{tan}x)}+\frac{1}{1-\mathrm{tan}x})}^{2}\phantom{\rule{negativethinmathspace}{0ex}}\mathrm{d}x=3\mathrm{ln}2-\frac{4}{\pi}G-\frac{1}{2},$

where $G$ is Catalan's constant.

I can see that replicating the techniques for the following integral could be rather lenghty.

${\int}_{0}^{\phantom{\rule{negativethinmathspace}{0ex}}\frac{\pi}{2}}\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}{(\frac{1}{\mathrm{log}(\mathrm{tan}x)}+\frac{1}{1-\mathrm{tan}x})}^{3}\phantom{\rule{negativethinmathspace}{0ex}}\mathrm{d}x$

My question: Could someone have, ideally, a different idea to evaluate the latter integral?

Logarithms
Answered

Chelsea Lamb
2022-09-27

For estimates, the inequality $\mathrm{log}(y)\le y-1,$$y>0$ is often helpful. Is there any sort of upper bound for the logarithm function in the complex plane? Specifically, $|\mathrm{log}(z)|\le $ something for all $z\in \mathbb{C}$

Perhaps this would work?: $\mathrm{log}(z)\le \sqrt{{\mathrm{log}}^{2}|z|+\mathrm{arg}(z{)}^{2}}$

Logarithms
Answered

Haiphongum
2022-09-26

I'm studying logarithms and I encountered this equation:

$[{\mathrm{log}}_{9}(k+1){]}^{2}+{\mathrm{log}}_{9}(k+1)+(k+1)>3$

I tried a lot but I still couldn't solve it! I know this may be easy for most of you but please could you help me?

Thanks!

Logarithms
Answered

HypeMyday3m
2022-09-25

I can only solve halfway through.

And why is

${10}^{\mathrm{log}(x)}=x$

Thanks

Logarithms
Answered

Daniella Reyes
2022-09-25

Our teacher has shown us examples for the same base and when it's both add and subtract. But I'm not sure how to do this.

Logarithms
Answered

zaviknuogg
2022-09-24

I've trying to solve this for quite a while now, but not being able to finish the proof.

Prove using induction that $(\mathrm{log}n{)}^{2}\le {2}^{n}$

Logarithms
Answered

Kaila Branch
2022-09-24

The given equation is

${x}^{\frac{3}{4}({\mathrm{log}}_{2}x{)}^{2}+{\mathrm{log}}_{2}x-\frac{5}{4}}=\sqrt{2}$

I took ${\mathrm{log}}_{2}x$

and then rewrote the given equation as

${x}^{3{t}^{2}+4t-5}=\sqrt{2}$

But I don't know what to do after this. How will I find the nature and no. of roots?

Logarithms
Answered

Megan Herman
2022-09-24

Can someone please explain what ${}^{2}\mathrm{log}x$ means? Is it the same as saying $\mathrm{log}{x}^{2}$ or is it something completely different?

Logarithms
Answered

Kaila Branch
2022-09-24

How can I evaluate following logarithmic integral:

$\underset{0}{\overset{1}{\int}}\frac{\mathrm{ln}x\mathrm{ln}(1-zx)}{1-x}dx$

Logarithms
Answered

hotonglamoz
2022-09-24

I'm studying logarithms and am doing an exercise where you're supposed to evaluate the solutions of common logarithms without using a calculator. I'm very stuck on this one particular question. I know the answer because I used my calculator, but I'd like to know how to solve it without one. The question is

$\mathrm{log}\left(\frac{10}{\sqrt[3]{10}}\right)$

How do I solve this without a calculator? (Please provide a step-by-step solution, this has really confused me.)

Logarithms
Answered

HypeMyday3m
2022-09-24

It's easy to prove that following upperbound is true:

$\sum _{i=1}^{N}{a}_{i}\mathrm{ln}{a}_{i}\le A\mathrm{ln}A$, where $\sum _{i=1}^{N}{a}_{i}=A$ and ${a}_{i}\ge 1$

I'm wondering, is there stronger upperbound?

Logarithms
Answered

pilinyir1
2022-09-23

I had problems understanding how to solve

${6}^{-{\mathrm{log}}_{6}^{2}}$

Any help would be much appreciated.

Thanks!

Logarithms
Answered

Daniella Reyes
2022-09-23

I want to solve this equation:

$8{n}^{2}=64n{\mathrm{log}}_{\text{}2}(n)$

After some steps, I get to a point in which I believe, the only way to proceed is to apply something like Bolzano's or Newton's method to find a solution.

I get to: $n=8{\mathrm{log}}_{\text{}2}(n)$

Of course with big numbers applying Bolzano would be very tedious and this is why I want to ask you if there is an analytical way of solving this, not by approximations.

Thanks a lot!

Logarithms
Answered

Lyla Carson
2022-09-23

Is it hard to prove this identity:

$2\mathrm{log}(a)\mathrm{log}(b)=\mathrm{log}(ab{)}^{2}-\mathrm{log}(a{)}^{2}-\mathrm{log}(b{)}^{2}$

for $a>1$ and $b>1$?

Logarithms
Answered

easternerjx
2022-09-23

$-{\mathrm{log}}_{8}2+5{\mathrm{log}}_{8}2+\frac{1}{2}{\mathrm{log}}_{8}(16)$

Logarithms
Answered

Sara Fleming
2022-09-23

$\sqrt{{\mathrm{log}}_{x}\left(\sqrt{3x}\right)}\cdot {\mathrm{log}}_{3}x=-1$

I am not entirely sure how to go about solving for $x$. I cannot square each side because the product isn't $\ge 0$, I can't think of any more approaches right now.

Logarithms
Answered

mangicele4s
2022-09-23

${\mathrm{log}}_{4}32=2.5$

If

${\mathrm{log}}_{a}(b\cdot c)={\mathrm{log}}_{a}b+{\mathrm{log}}_{a}c\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}};(a>0,b>0,c>0,a\ne 1)$

Then why does ${\mathrm{log}}_{4}32$ can't be ${\mathrm{log}}_{4}(4\cdot 8)={\mathrm{log}}_{4}4+{\mathrm{log}}_{4}8=1+2=3$?

Logarithms
Answered

easternerjx
2022-09-23

One can use Taylor series of the log or exp function to get the result that $x=1$. I was wondering if there is any other simple solutions.

Thanks a lot!

Logarithms
Answered

Alexus Deleon
2022-09-23

I've been thinking about a natural number like $n$ so that ${x}^{x}=n$ for some irrational $x$ but i couldn't find anything. As i didn't know how to approach the problem at all, i tried to make some simpler cases first ($n$ is a natural number):

1.${\sqrt{a}}^{\sqrt{a}}=n$ for irrational $\sqrt{a}$

My approach: ${\sqrt{a}}^{\sqrt{a}}=n\Rightarrow {\sqrt{a}}^{a}={n}^{\sqrt{a}}$. And here, We suppose $a$ is even. This means the LHS would be a natural number, and thus ${n}^{\sqrt{a}}$ is natural too. This means $\sqrt{a}={\mathrm{log}}_{n}b$ which i think can not be true when $b$ is not a power of $n$ because that time i think the logarithm would be transcendental (i'm not sure). But this only disproves the case for $a$ being even! Still if we can prove theres no such $n$ and $a$, we should check the next case.

2.${a}^{a}=n$ for algebraic and irrational $a$

This seems more likely than the first case, but still i have no idea in approaching it.

3.${a}^{a}=n$ for irrational $a$

This is indeed more general than the first two cases and we should check it if the first two cases failed! Although it may be great to find all kinds of irrational $a$s so that the equality will hold for natural $n$

I would appreciate any help :)

Logarithms
Answered

beobachtereb
2022-09-21

How do I simplify this? This is what I have done so far:

$2{\mathrm{log}}_{3}5={\mathrm{log}}_{3}{5}^{2}={\mathrm{log}}_{3}(25)$

${3}^{{\mathrm{log}}_{3}(25)}$

What do I do from here? And the answer is one of these mixed solutions:

$0$

$-2$

$-\frac{\pi}{4}$

$\frac{1}{x+2}$

$\pm \frac{4}{25}$

$25$

$30\xb0$

$2$

$3$

$5$

$\pi $

$\frac{\pi}{3}$

$(-\mathrm{\infty},2)$

$4(x+1{)}^{2}+3$

$-\frac{\sqrt{2}}{2}$

$-\frac{\sqrt{3}}{2}$

$\frac{\sqrt{2}}{2}$

Finding logarithms questions and answers online is always pure luck, especially if these are provided with detailed answers to questions. This is exactly what you can find below regardless of your practical application, college course, or complexity. These logarithms questions are related to higher education courses, which is why they will help you find the answers to what you currently have. When you deal with logarithmic equations you are approaching an exponential equation where the variable appears in an exponent as it’s in (log b (bx) = x. It also shows that the logarithmic graph equation can be represented as well.