Questions tagged [logarithms]

Questions related to real and complex logarithms.

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Analytical way to solve $a^x+b^{x^2}=c$?

somewhere on internet I saw someone asking if it's posisble to solve $2^x+3^{x^2}=6$ analytically. If it was $\times$ and not + it would be pretty easy, but here, I have no idea.
• 1,157
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What's the opposite of a lognormal distribution?

If $Y$ is normally distributed, and $X = \exp{(Y)}$ then $X$ has a log-normal distribution. I'm curious about the opposite case. If the pdf of $Y$ is a half-normal distribution (normally distributed ...
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How can I draw $f(S)$ with $f(z)=z^{1+i}$ where $S=\{0<Arg(z)<\pi/6\}$?

Let $f:\Bbb{C}\setminus(-\infty, 0]\rightarrow \Bbb{C}$ where $f(z)=z^{1+i}:=e^{(1+i)\log(z)}$. We consider $$S=\{0<Arg(z)<\pi/6\}$$ I want to draw $f(S)$. If I take $z\in S$ then $z=re^{it}$ ...
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• 529
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Branch cut of $\operatorname{Log}(z^2-z)$

$\newcommand{\Arctan}{\operatorname{Arctan}}\newcommand{\Ln}{\operatorname{Ln}}\newcommand{\Arg}{\operatorname{Arg}}\newcommand{\Log}{\operatorname{Log}}$We have $\Log(z^2-z)=\Ln|z^2-z|+i\Arg(z^2-z)$. ...
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• 5,384
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Derive an equation of the form Y = MX + C from $y\:=\:px^2+q\sqrt{x}$, where p and q are constants

Hello and Good day to you all. I have been trying to linearize the following equation to the form Y = MX + C in order to plot a straight a line graph with a given set of x and y values. I have arrived ...
• 135
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Proof that the inverse of $f(x)={10}^x$ is $f^{-1}\left(x\right)={\mathrm{log} x\ }$.

I know that the inverse of $f(x)={10}^x$ is $f^{-1}\left(x\right)={\mathrm{log} x\ }$, geometrically this implies reflecting the original function across the line y=x. I am also aware that this can be ...
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Logarithm of a positive-definite matrix

If $z$ is a nonzero complex number, then we can write it in the form $z=e^w$ where $w$ is another complex number which is not unique. For example we can do $z=re^{i\theta}=e^{\ln(r)+i(\theta+2k\pi)}$, ...
• 2,480
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Can I further simplify $\frac{-2\log 6}{\log 6-\log 4}$?

Can I further simplify this logarithmic expression somehow in this case, or is it already the point when I take a calculator?
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Visualization of complex exponential and logarithmic map as $z$-plane to $\omega$-plane map.

I am studying complex analysis and themain thing about complex analysis that disturbs me is the difficulty in visualization.I have studied exponential function and logarithmic function in complex ...
• 1,625
106 views

Solve $x^{2^{\sqrt{2}}} = {\sqrt{2}}^{2^x}$

How to solve: $$x^{2^{\sqrt{2}}} = {\sqrt{2}}^{2^x}$$ where $x \in R^{+}$? We take log based on 2 on both sides, then $2^{\sqrt{2}} \log_2 x = 2^x \log_2 {\sqrt{2}} = 2^{x-1}$ (thanks for the comment ...
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What would be a good metric that puts heavier weight on a longer time period?

I'm currently trying to think of a metric to measure a performance of a product that gives higher points to longer contracts. For example, let's say that one contract A is a year long contract and the ...
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If the value of $\log 2 = 0.3010$, then the number of zeroes after the decimal point till the first non-zero digit in $0.16^{20}$ is?

I had noted down this statement in my notebook which says that "If $-n$ is the characteristic of $\log_{10}y$, then the number of zeroes between the decimal and the first significant digit after ...
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Prove that $$\ln (1+z+o(z))=z-\frac{z^{2}}{2}+o\left(z^{2}\right), \quad z \rightarrow 0.$$ My attempt is to use Taylor's expansion, so we get $$\ln (1+z+o(z))=z+o(z)+o(|z+o(z)|^2).$$ What are ...