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.
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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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If $\log_a3=p$ and $\log_a2=q$ what strategies deduce an expression for $\log_a(4.5a^2)$?

If $\log_a3=p$ and $\log_a2=q$ what strategies deduce an expression for $\log_a(4.5a^2)$? I've considered the exponent forms $a^p=3$, $a^q=2$ and $a^x=4.5a^2$ and other qualities of logs such as $\...
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When two functions touch

I have two functions: $$\log P=e^{-2\pi\xi_{o}^{2}t}\frac{t}{\tau}$$ and $$\log P=\left(E_{0}^{2}\tau t\right)^{-2}\frac{t}{\tau}$$ And they touch at some point, there's a transition of the first ...
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Justification for this Calculus "Shortcut" (Logarithm of Derivatives)?

In school, we were shown a "shortcut" (I am not sure if this "shortcut" even has a name) in calculus that describes the relationship between the derivatives of a function, but we ...
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Show that $\lim_{z \to 0 }\frac{\log(z+1)}{z} = 1$ for the complex logarithm

Show that $\lim_{z \to 1 }\frac{\log(z)}{z} = 1$ for the complex logarithm, with the definition of the complex logarithm being $$ \log(z) = \log |z| + i \arg(z). $$ Edit. Ok, so it seems that it was ...
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Laplace transform of natural logarithm of a function

Let's assume that we have a function $i(t)$ and it has a Laplace transform $I(s)$. Can we calculate the Laplace transform of $\ln(i(t))$ in terms of $I(s)$ ?
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$\log(x)+\log(y)$ with respect to $x+y$

This is probably a very dumb question, but say that we have the quantity $(x+y)$ which we can access directly, not knowing what $x$ and $y$ is separately. Can we obtain an expression for $\log(x)+\log(...
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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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What happens if I apply the complex powerfunction to a unbounded sector of the complex plane?

I have the following definition of the power function: Let $\Omega\subset \Bbb{C}$ be simpy connected such that $0\notin \Omega$. Then for $\alpha\in \Bbb{C}$ and $z\in \Omega$ we define $$z^\alpha:=...
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Computing the log of a sum of exponentials

in a Coursera course by UW I've come across this piece of code computing the log of a sum of exponentials. ...
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Proof of logarithmic asymptotic identity

I saw the following identity here: $$ \log(a\,b) = \Theta(\log(a + b)) = c\log(a + b), $$ for any $a,b>1$ and some constant $c>0$ independent of $a,b$. Could someone provide any hints why this ...
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Is there a transform to divide a log by another log with the same base such that the original logs do not need to be evaluated?

It’s been suggested to me that I can solve for x in $5^x$$^-$$^1$$ = 2$ by converting to log base 10. I’ve tried converting to base 10 but I arrive at $x=\frac{\log2}{\log 5}+1$ and am unable to go ...
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How to approxiamte the following integral related to the Modified Bessel function of the first kind?

Maybe I could approximate the following integral related to the modified Bessel function of the first kind ? $$\mathbb{E}_{Z_{1},Z_{2},\hat{h}} \log \frac{I_{n_{\mathrm{d}}-1}\left(2 \left|\hat{h}\...
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How do you get an expression for $x$ in terms of $c$ in the equation $\ln(x)=cx$?

I am in the middle of solving this problem and I came across a sub problem that sorta has the format $$\ln(x)=cx$$ I tried looking online on how to solve for $x$ if given $c$ but I couldn't find ...
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Can the mantissa of a log ever be zero?

I had a doubt today when our teacher told us this: A logarithm of the form log N can be written as n + f, where n is an integer known as the characteristic, and f is a fraction where 0 < f < 1. ...
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209 views

$F$ such that $F(A_{0}B_{0}+A_{1}B_{1})=F(A_{0})+F(B_{0})+F(A_{1})+F(B_{1})$?

We know that the only way of turning a multiplication into addition is by a logarithmic function; so if $$C=A_{0}B_{0}$$ then we know that: $$\log(C)=\log(A_{0})+\log(B_{0})$$ but can we extend this ...
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Mellin transformation of Log in denominators

Does anyone know how to compute this integral? $ \int_0^1d r \int_0^1d h\ h^{x-1}r^{y -1}\left\lbrace \frac{1}{z-w\left(\ln \frac{rh}{(1-h)(1-r^2h)} \right) }\right\rbrace$ This could interpret as a ...
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Prove the logarithm property $\ln(\frac 1a) = -\ln(a)$ using its definition of fundamental theorem of calculus (FTC).

I know how to prove that. But I have a question about variable substitution. Problem statement: Prove directly from the definition $ L(x) = \int_{1}^x \frac 1tdt $ that $L\left(\frac 1a\right) = -L(a)...
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Help me with this integral

Find the value of $$\int\frac{1+\ln x}{4+x\ln x^2}\mathrm{d}x$$ I have a very bad understanding of integrals where some function of a variable is in the denominator. I know I have to do some kind of ...
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Could $\log_1(1)$ have more than one value?

I was wondering if $\log_1(1)$ could have more than one value, due to the fact that in the equation $1^x = 1$, $x$ can be $0$ and $1$. Thanks in advance.
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Proving $ e^2 = e$? [closed]

Given the function, $$ f(x)= \frac{1}{e^x-1} \prod_{n=1}^{\infty} \frac{x^n}{n!} $$ Then applying the log, $$ \ln(f(x)) = \frac{1}{e^x-1} \sum_{n=1}^{\infty} \frac{x^n}{n!} = \frac{e^x-1}{e^x-1} = ...
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Is there a mapping function to transform $m^n \to \sum_{i \in K}{2^i}$ (any power to sums of powers of 2).

From the idea of sums of powers of 2 described in this question is very clear that we can form any natural $\Bbb N$ from a sum of powers of 2. So what would be, the transformation from: $$a = m^n \to \...
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An intuitive explanation or a proof of $n^{log_a b} = b^{log_a n}$

I am reviewing logarithms and ran into this relationship $n^{log_a b} = b^{log_a n}$ Could use some help understanding it
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How many integer values of $y\in(-2022;2022)$ are there such that there exists $x$ wherein $2\log_2(x+y\sqrt{3})-2=\log_{\sqrt{3}}(x^2+y^2-1)$?

How many integer values of $y \in (-2022; 2022)$ are there such that there exists $x \in \mathbb R$ where the following condition is satisfied? $$2\log_2(x + y\sqrt{3}) - 2 = \log_{\sqrt{3}}(x^2 + y^2 ...
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-4 votes
1 answer
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Let a and b be positive integers such that $a + b = 10.$ The number $a^{10}b^{2022}$ represented as a decimal has 618 digits. Find $a, b$. [closed]

Let a and b be positive integers such that a + b = 10. The number $a^{10}b^{2022}$ represented as a decimal has 618 digits. Find a and b. Honestly, when I came across this question, I couldn't figure ...
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1 answer
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Log rules for calculating joint entropy

This question is probably not so hard for you. Why is the entropy equal to: $$ H(x,y)=2\log_2(5)-\frac{8}{25}\log_2(2)-\frac{6}{25}\log_2(3), $$ for the following joint distribution? $$ p(x,y) = \...
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15 votes
2 answers
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Does there exist a function which converts exponentiation into addition?

A useful property of the logarithm is that it can "convert" multiplication into addition, as in $\ln(a)+\ln(b)=\ln(ab) \text{ for all } a, b \in \mathbb{R}^+$ Does there exist a function $f$,...
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Domain of $h(x) = \log_2 (x^2+4)$

I am setting the $x^2+4 > 0$ and solving, but this leaves me with a question I can't seem to understand. The resource I'm using is saying the domain is $(-\infty, \infty)$; $x$ is a set of all Real ...
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Is $n+1<n^{\frac{n-6}{n-7}}$ equivalent to $\frac{n-6}{n-7}\gt\log_{n}{(n+1)}$?

When I have inequality for $n\in\mathbb{N}\wedge n\ge11$ such as: $n+1<n^{\frac{n-6}{n-7}}$. Can I turn it into a logarithmic inequality: $\;\frac{n-6}{n-7} \gt \log_{n}{(n+1)}$ ? Will it give me a ...
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Exponential sum behaves like linear term for large $t$

I've done some calculations on interesting mathematical objects and came to the conclusion that they would behave nicely as expected if we would have that $$t \sim 2\sum_{n=1}^\infty \exp(-\pi n^2/t^...
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which log operations are involved in this formula?

I've a function f1(x) that "scale" (not sure how to call it) a power of 2 function, such as: Now, from this value, I display the dB measure (in the form ...
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In a model why can we write $-\frac{x_2}{x_1}d\left(\frac{x_1}{x_2}\right) = -\left(\frac{dx_1}{x_1} - \frac{dx_2}{x_2}\right)$

As above, the model is non-stochastic, not exactly sure if that has any implications. Would it be possible to take the fact that $\frac{d(\ln x)}{dx} = \frac{1}{x}\to\frac{dx}{x} = d(\ln x)$, and ...
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2 votes
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Evaluating $b=1+2^\sqrt{\log_23}-3^\sqrt{\log_23}$

Evaluating $b=1+2^\sqrt{\log_23}-3^\sqrt{\log_23}$ I am aware that $a^{\log_ax}=x$ but that doesn't seem to apply here. $b=1+e^{\ln 2^\sqrt{\log_23}}-e^{\ln3^\sqrt{\log_23}}=1+e^{\sqrt{\frac{\ln3}{\...
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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 ...
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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 $z=re^{i\theta}=e^{\ln(r)+i(\theta+2k\pi)}$, ...
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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 ...
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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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Expansion of logarithm with little o

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 ...
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Bounding the difference between two points of the logarithm

Let $n$ and $k$ be positive integers. Is it possible to obtain an asymptotic upper bound on the following difference: $$ \log(n) - \log(n-k) $$ Looking at the output of WolframAlpha it seems that, if $...
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Inverse of $\frac{\log(a-x)}{\log(x)}$

I was working with a series, and found that my problem include this function $f(x)=\dfrac{\log(a-x)}{\log(x)}$? Does anyone know what the inverse of this function is?
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-2 votes
1 answer
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Need help to understand the steps to the solution $(2/3)^x=6/25$ using complex numbers

Trying to understand the specific steps of the solution to $(2/3)^x=6/25$ whose solution is $$x = {\log(25/6) - 2i\pi n \over\log(3/2)}$$ with $n \in \mathbb Z$ and real solution $$x= {\log(2)+\log(3)-...
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2 votes
1 answer
41 views

Lambert W of a constant multiple

$ \ln(cx) $ can be expressed as $ f(c) + \ln(x) $, where $ f(c)=\ln(c) $. Does the lambert W function have a similar property? (Can $ W(cx) $ be expressed as $ f(c)+W(x) $ for some function $ f $).
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-1 votes
1 answer
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Where is the base in this logarithm? [closed]

We are studying logarithms in our math class and I stumbled upon this task: I am not asking for the solution of the whole problem, but, to tell you the truth, I don't understand anything in it! It ...
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Logarithmic equation with $\lg^2$

$$\lg^210x+\lg x=19$$ Could anyone please help me to understand what exactly is squared in this equation? The whole logarithmic function? Some kind of example with numbers would be super helpful. ...
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