All Questions
Tagged with algebra-precalculus logarithms
1,541
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Compute a tight upper bound of $\sum_{i=1}^{n-1}\frac{1}{3^i\log{n}- 3i}$?
I am trying to compute a tight upper bound of the sum below.
$\sum_{i=1}^{n-1}n\frac{\frac{1}{3^i}}{\log_3{(n/3^i)}}$
I was able to 'simplify' it up to the expression below.
$n\sum_{i=1}^{n-1}\frac{1}{...
- 185
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1
answer
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Showing $ (2/3)^{\log_{3/2}(n/n_0)} =n_0/n $
I read the following identity in the CLRS Introduction to Algorithms book, and I can't work out the computation.
$$
(2/3)^{\log_{3/2}(n/n_0)} =n_0/n
$$
I tried to expand the exponent using the ...
- 185
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Does $\log\left(\frac{A}{B}\right)$ really equal to $\log(A)-\log(B)$? [duplicate]
I was investigating the laws of logarithm and playing with Desmos when I realized something curious.
The example equation is $f(x)=\log\left(\frac{2x-4}{6x-8}\right)$ and the graph is this:
The law ...
- 11
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Does this equation have a closed-form solution for $f(x) = 0$?
I have the equation $f(x) = -n^{-x} × (n + 1)^{x - n - 2} × \left(\left((n + 1)^{n + 1} - n^n × x\right) × (\ln(n) - \ln(n + 1)) + n^n\right)$ where $n$ is a positive real number and $\ln(z)$ is the ...
- 1,861
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3
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Can the "simple" equation $e^x=\log(x)$ be solved using algebra?
I came across this really simple-looking yet astonishingly hard problem to solve:
$$e^x=\log(x).$$
I tried to use Lambert-W function, but I cannot get it to the required standard form. Even Wolfram ...
- 13
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3
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How do you find the exact value of a logarithm with a radical in the base?
I'm struggling to find a method for evaluating $\log_{5\sqrt2} 50$ (or ${\log50}\over{\log5\sqrt2}$) without using a calculator. When using a calculator, I am given an exact value of 2, but I can't ...
1
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3
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325
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How do I solve $x^{4^x}=4$?
My friend showed me this problem from Twitter and I am struggling to solve it. I see that I can manipulate it into several equations (some of which I'll insert below), but none seem to be any progress ...
- 1,688
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Simplifying the expression $\frac{1}{3} \ln (x+2)^3+\frac{1}{2}\left[\ln x-\ln \left(x^2+3 x+2\right)^2\right]$ [closed]
Express as a single logarithm. Simplify.
$$
\frac{1}{3} \ln (x+2)^3+\frac{1}{2}\left[\ln x-\ln \left(x^2+3 x+2\right)^2\right]
$$
So I am posting the question, how I solved it and then how the TA ...
- 137
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1
answer
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Calculating (Approximate) values of fractional powered real numbers without calculator or log/antilog tables.
Is there any way to calculate/approximate values like $$(\frac{125}{250})^{0.66}$$ using only a pen, paper and the mind?
(Above expression being just an example, the numbers may vary and not be easy ...
- 35
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5
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595
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Prove that $\ln x\leq\frac{x^{x+1/x}-1}{2}$ [ not solved ]
Prove that $$\ln x\leq\frac{x^{x+1/x}-1}{2}$$ is true for every positive real number, without calculus/derivative . (i.e. using some inequalities)
My progress. For $x\geq 1$ using $x+1/x\geq 2$ we ...
- 65
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2
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Prove that $\ln x\leq\frac{x^{\frac{x+\sqrt{x}+1}{2\sqrt{x}}}-1}{\sqrt{x}}$ for all positive real numbers
I am trying to prove the inequality $$\ln x\leq\frac{x^{\frac{x+\sqrt{x}+1}{2\sqrt{x}}}-1}{\sqrt{x}}$$ is true for all positive real numbers, without using calculus.
I realized the equality occurs if ...
- 65
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2
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133
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Does the second positive solution (besides $x = 1$) of the equation $e^{x^2-1}=x^3-x\ln x$ have a closed form?
Does the equation $$e^{x^2-1}=x^3-x\ln x$$ have a closed form solution ?
The given equation has $2$ positive real roots. Graphically
It is not hard to see that $x=1$ is a rational solution. The ...
- 644
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How to solve the equation $x^{\log_3{(x-1)}} = 3^{(3+\log_3{x})}$ [SOLVED]
I made a mistake copying the question ! it should be $x^{\log_3{(x)}-1} = 3^{(3+\log_3{x})}$.
I'm trying to solve the equation:
$x^{\log_3{(x-1)}} = 3^{(3+\log_3{x})}$
What I tried:
I took the ...
- 23
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51
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Question about $\log_1 (1)$ [duplicate]
In my college calculus class we just covered properties of logs, and I wanted to ask about them. Two of them are these:
For all $0 < a$: $$\log_a1=0$$
and for all $-\infty\leq b \leq \infty$: $$\...
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Simplifying inverse log-log transformation of a rational function
Consider a rational function fit of a function after a log-log transformation:
$\hat{y} = f(\hat{x}) = \frac{n(\hat{x})}{d(\hat{x})}$,
where $\hat{x} = \log_{10}(x)$ and $\hat{y} = \log_{10}(y)$. For ...
