 # Logarithms questions and answers

Recent questions in Logarithms Conrad Beltran 2022-09-27

### 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$\mathrm{log}\left(\frac{10}{\sqrt{10}}\right)$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

### Solving $\mathrm{log}\left(x\right)=x-1$?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! Alexus Deleon 2022-09-23
### Irrational to power of itself is naturalI'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: ${\sqrt{a}}^{\sqrt{a}}=n⇒{\sqrt{a}}^{a}={n}^{\sqrt{a}}$. 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 :) beobachtereb 2022-09-21