Questions tagged [logarithms]
Questions related to real and complex logarithms.
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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)...
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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 $\cos x=\sin x$ ...
user1316790
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Derivative of $e^{x+e^{x+e^{x+...}}}$
Let $y=$ $e^{x+e^{x+e^{x+...}}}$
To find $\frac{dy}{dx}$, I took the natural log on both sides, which gives: $$\ln y = x + e^{x+e^{x+e^{x+...}}}$$
Differentiating on both sides,$$\frac{1}{y}\frac{dy}{...
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How to prove upper bound of this difference of the Sine Integral?
This exercise can be found in Mathematics LibreTexts (bottom of the page) . I have been stuck for about a day and have made minimal progress.
Let $S(x)=\int_0^x\frac{\sin t}{t}$.
Show that for $k \ge ...
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Logarithmic Function Calculation in Mathematica
I find these results in the evaluation of the logarithms that only differ in the sign $-$ I do not understand why in the first case $\operatorname{Log}[x+1]/8$ is not returned as an answer.
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Construction of discontinuous $f$ such that $f(xy) = f(x)+f(y)$ [duplicate]
Question
How to construct a discontinuous $f$ such that $f(xy) = f(x)+f(y)$. Domain of $f$ has to be some subset of $\mathbb{R}$ and range of $f$ is $\mathbb{R}$. Also, try to construct non ...
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Log X to what base n yields a whole number [closed]
Does there always exist a real number 'n' such that $log_{n}x$ is a whole number for any real number x?
If yes what would the function to find this number look like?
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Why is there no logarithmic form of the exponential distributive rule/power of a product rule?
When learning the laws of exponents and logarithms, one finds that there is a correspondence. Each law of exponents has a corresponding equivalent expression in terms of logarithms. For example, the ...
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Which one is closer to $3024^{2500}$? $10^{8000}$ or $10^{9000}$?
I first approached this question by applying log to $3024^{2500}$.
$\log(3024^{2500}) = 8701.454467\cdots$
I then thought that since $8701$ is closer to $9000$, $3024^{2500}$ is closer to $10^{9000}$...
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When does $x\ln f(x)$ become convex?
When a function $f>0$ is defined on $x\geq 0$, I would like to know the conditions for $F(x)=x\ln f(x)$ to be convex.
Naively, $f$ being convex looks sufficient, but it is not true even if $f$ is ...
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Euler Sums of Weight 6
For the past couple of days I have been looking at Euler Sums, and I happened upon this particular one:
$$
\sum_{n=1}^{\infty}\left(-1\right)^{n}\,
\frac{H_{n}}{n^{5}}
$$
I think most people realize ...
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Implicit function equation $f(x) + \log(f(x)) = x$
Is there a function $f \colon \mathbb{R}_{>0} \to \mathbb{R}_{>0}$ such that
$$
f(x) + \log(f(x)) = x
$$
for all $x \in \mathbb{R}_{>0}$?
I have tried rewriting it as a differential equation ...
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Natural Log's Property Doesn't Transfer Over
I am trying to rewrite the summation of $\ln(x)$ equation into a continuous function using logarithmic properties. We already know that $\left(\sum_{n=1}^{x}\ln\left(n\right)\right)$ is just equal to $...
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Is ln|x| + C really the most general antiderivative of 1/x? [duplicate]
I recently stumbled across a claim that $\ln |x| + C$ isn't the most general antiderivative of $1/x$. The argument was that the parts of the curve $\ln |x|$ separated by the $y$-axis do not have to be ...
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