Recent questions in Logarithms

Evaluating ${\int }_{0}^{\frac{\pi }{2}}{\left(\frac{1}{\mathrm{log}\left(\mathrm{tan}x\right)}+\frac{1}{1-\mathrm{tan}\left(x\right)}\right)}^{3}dx$Using the method shown here, I have found the following closed form.${\int }_{0}^{\phantom{\rule{negativethinmathspace}{0ex}}\frac{\pi }{2}}\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}{\left(\frac{1}{\mathrm{log}\left(\mathrm{tan}x\right)}+\frac{1}{1-\mathrm{tan}x}\right)}^{2}\phantom{\rule{negativethinmathspace}{0ex}}\mathrm{d}x=3\mathrm{ln}2-\frac{4}{\pi }G-\frac{1}{2},$where $G$ is Catalan's constant.I can see that replicating the techniques for the following integral could be rather lenghty.${\int }_{0}^{\phantom{\rule{negativethinmathspace}{0ex}}\frac{\pi }{2}}\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}{\left(\frac{1}{\mathrm{log}\left(\mathrm{tan}x\right)}+\frac{1}{1-\mathrm{tan}x}\right)}^{3}\phantom{\rule{negativethinmathspace}{0ex}}\mathrm{d}x$My question: Could someone have, ideally, a different idea to evaluate the latter integral?

Chelsea Lamb 2022-09-27

Does the logarithm inequality extend to the complex plane?For estimates, the inequality $\mathrm{log}\left(y\right)\le y-1,$$y>0$ is often helpful. Is there any sort of upper bound for the logarithm function in the complex plane? Specifically, $|\mathrm{log}\left(z\right)|\le$ something for all $z\in \mathbb{C}$Perhaps this would work?: $\mathrm{log}\left(z\right)\le \sqrt{{\mathrm{log}}^{2}|z|+\mathrm{arg}\left(z{\right)}^{2}}$

Haiphongum 2022-09-26

Logarithmic equationI'm studying logarithms and I encountered this equation:$\left[{\mathrm{log}}_{9}\left(k+1\right){\right]}^{2}+{\mathrm{log}}_{9}\left(k+1\right)+\left(k+1\right)>3$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!

HypeMyday3m 2022-09-25

Can one use logarithms to solve the equations $2={3}^{x}+x$ and $2={3}^{x}x$?I can only solve halfway through.And why is${10}^{\mathrm{log}\left(x\right)}=x$Thanks

Daniella Reyes 2022-09-25

Express $4\mathrm{ln}\left(x\right)+2\mathrm{ln}\left({x}^{4}{y}^{3}\right)+5\mathrm{ln}\left(z\right)$ as a single logarithmOur 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.

zaviknuogg 2022-09-24

Prove $\left(\mathrm{log}n{\right)}^{2}\le {2}^{n}$ by inductionI've trying to solve this for quite a while now, but not being able to finish the proof.Prove using induction that $\left(\mathrm{log}n{\right)}^{2}\le {2}^{n}$

Kaila Branch 2022-09-24

no. and nature of roots of ${x}^{\frac{3}{4}\left({\mathrm{log}}_{2}x{\right)}^{2}+{\mathrm{log}}_{2}x-\frac{5}{4}}=\sqrt{2}$The given equation is${x}^{\frac{3}{4}\left({\mathrm{log}}_{2}x{\right)}^{2}+{\mathrm{log}}_{2}x-\frac{5}{4}}=\sqrt{2}$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?

Megan Herman 2022-09-24

What does this log notation mean?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?

Kaila Branch 2022-09-24

Integral $\underset{0}{\overset{1}{\int }}\frac{\mathrm{ln}x\mathrm{ln}\left(1-zx\right)}{1-x}dx$How can I evaluate following logarithmic integral:$\underset{0}{\overset{1}{\int }}\frac{\mathrm{ln}x\mathrm{ln}\left(1-zx\right)}{1-x}dx$

hotonglamoz 2022-09-24

Common logarithm questionI'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.)

HypeMyday3m 2022-09-24

Upperbound for $\sum _{i=1}^{N}{a}_{i}\mathrm{ln}{a}_{i}$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?

pilinyir1 2022-09-23

Logarithmic equations with different basesI had problems understanding how to solve${6}^{-{\mathrm{log}}_{6}^{2}}$Any help would be much appreciated.Thanks!

Daniella Reyes 2022-09-23

How to solve this logarithmic equation?I want to solve this equation: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: 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!

Lyla Carson 2022-09-23

Product of logarithms, prove this identity.Is it hard to prove this identity:$2\mathrm{log}\left(a\right)\mathrm{log}\left(b\right)=\mathrm{log}\left(ab{\right)}^{2}-\mathrm{log}\left(a{\right)}^{2}-\mathrm{log}\left(b{\right)}^{2}$for $a>1$ and $b>1$?

easternerjx 2022-09-23

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

Sara Fleming 2022-09-23

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

mangicele4s 2022-09-23

Why does ${\mathrm{log}}_{4}32\ne {\mathrm{log}}_{4}\left(4\cdot 8\right)$${\mathrm{log}}_{4}32=2.5$If${\mathrm{log}}_{a}\left(b\cdot c\right)={\mathrm{log}}_{a}b+{\mathrm{log}}_{a}c\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}};\left(a>0,b>0,c>0,a\ne 1\right)$Then why does ${\mathrm{log}}_{4}32$ can't be ${\mathrm{log}}_{4}\left(4\cdot 8\right)={\mathrm{log}}_{4}4+{\mathrm{log}}_{4}8=1+2=3$?

easternerjx 2022-09-23