# Questions tagged [logarithms]

Questions related to real and complex logarithms.

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### What is the product of the solutions to the equation (√10)(x^(log (x))) = x^2? [closed]

I have spent a few hours on this question but I can't seem to grasp it. I have found that taking the log of both sides doesn't give me any progress. The log exponent identity didn't work. I also tried ...
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### Prove that sequence $(a_n)_{n\geq 1}$ is not convergent.

The standard branch of logarithm, $\log:\mathbb{C}\setminus (-\infty,0]\to\mathbb{C}$ is defined as $$\log(z):=\ln|z|+i\operatorname{Arg}(z)\tag{2}$$ where $\arg(z)$ is the standard branch of the ...
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### Integral of Complex logarithm makes sense?

For example, I know that the principal branch of logarithm is not defined over negative real axis. I think the integral of this logarithm along a circle doesn’t make sense. Moreover, I know that the ...
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### What does "any polynomial dominates any logarithm" mean here?

My textbook states that any polynomial dominates any logarithm: $n$ dominates $(\log n)^3$. This also means, for example, that $n^2$ dominates $n\log n$ However, it wasn't clear to me what the ...
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### Logarithms find the solution

What conditions must $a$ and $b$ satisfy for the equation to have at least one real solution? Find all the solutions of this equation: $1+\log_b(2\log(a)-x)\log_x(b)=2\log_b(x)$ I have tried ...
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### When Does ((n^a)-1)/a)) Equal e; A Sophomore's plight

I am a high school student (sophomore) and have come across something I would like explained. I was watching 3blue1brown for an explanation of calculus, when he used the formula: lim a->0 (d/dx(n^x)...
1 vote
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### What to consider when taking kth root on both sides of equality

Say I have the following expression: $10^{l} = a^{k}$ If I take the kth root of both sides, does that mean we get: $10^{\frac{l}{k}} = a$ We don't have to consider anything with plus or minus?
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### Solve the equation: $\log_{\sin x} (\cos x) - 2 \log_{\cos x} (\sin x) + 1 = 0.$ [closed]

Solve the equation: $\log_{\sin x} (\cos x) - 2 \log_{\cos x} (\sin x) + 1 = 0.$ Attempt: I transorm this equation in $(\log\cos x-\log\sin x)(\log\cos x+2\log\sin x)=0$, therefore $\cos x=\sin x$ ...
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### Inequality for log-convex functions

Let $s:\mathbb{R}^+\to\mathbb{R}^+$ be a positive, decreasing, log-convex function such that $s(0)=1$, $\lim_{x\to\infty}s(x) = 0$. In addition, $-s'$ is also assumed to be log-convex. Are these ...
I am pretty happy with the definition of the "maximal analytic contiuation of the logarithm" as  \operatorname{Log}_{\gamma}\left(z\right) = \int_\gamma ...
For example if I was differentiating $\ln(2x)$ doesn't the chain rule dictate that it should be $2/x$, not $1/x$? Why does the $2$ disappear?