All Questions
Tagged with logarithms calculus
1,378
questions
1
vote
1
answer
78
views
How to show $(\log_2 x)^4 \leq x^3$ for $x > 1$?
In Rosen's discrete Math textbook, they mention in the solutions for one problem that $(\log_2 x)^4 \leq x^3$ for $x > 1$. However, I'm not sure how to exactly derive that myself, nor does the ...
- 883
0
votes
6
answers
195
views
How would you prove $\log_{2}x < \sqrt x$ for $x > 16$? [closed]
I'm not really showing how to prove this, since I tried finding the $x$-intercepts/zeros of $f(x) = \sqrt x - \log_{2} x$ , and see that $x = 4, 16$ work but inspection, but I'm not sure how to ensure ...
- 883
3
votes
4
answers
165
views
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,$$\frac{1}{y}\frac{dy}{...
- 127
3
votes
2
answers
204
views
Why does the scalar inside a natural log dissapear when differentiating it? [closed]
For example if I was differentiating $\ln(2x)$ doesn't the chain rule dictate that it should be $2/x$, not $1/x$? Why does the $2$ disappear?
user1329181
8
votes
1
answer
276
views
Approximating $\log x$ by a sum of power functions $a x^b$
Let's approximate $\log x$ on the interval $(0,1)$ by a power function $a x^b$ to minimize the integral of the squared difference
$$\delta_0(a,b)=\int_0^1\left(\log x-a x^b\right)^2dx.\tag1$$
It's ...
- 47.3k
3
votes
1
answer
146
views
how to evaluate this integral $\int_0^1 \frac{\ln x \, \text{Li}_2(1-x)}{2+x} \, dx$
Question statement: how to evaluate this integral $$\int_0^1 \frac{\ln x \, \text{Li}_2(1-x)}{2+x} \, dx$$
I don't know if there is a closed form for this integral or not.
Here is my attempt to solve ...
1
vote
3
answers
115
views
How to solve $x+1=5e^{4x}$ [closed]
How to solve $x+1=5e^{4x}$
In general, I know to take ln() of both sides to bring down the exponent for e, but the left side is also a variable.
- 4,771
0
votes
2
answers
39
views
How do I move the origin of a logarithmic spiral?
Context: I'm a high school junior (in pre-calculus) and I'm trying to teach myself how to graph a logarithmic spiral for a math/art project.
Basically, I need to move the center of a spiral from the ...
-2
votes
1
answer
59
views
How does $\log(y)=C+t$ become $y = C e^{t}$? [closed]
I came across this transformation :
$$\begin{align}
\log(y) &= C + t \tag{1} \\[4pt]
y &= C e^{t} \tag{2}
\end{align}$$
How was the first step simplified into the second?
- 7
0
votes
3
answers
101
views
How does $3^{-\ln(n)}$ become $\frac{1}{n^{\ln(3)}}$? [closed]
$3^{-\ln(n)} =\frac{1}{n^{\ln(3)}}$.
I can't figure out how to get from the left side of the equation to the right. Any tips?
- 37
6
votes
0
answers
156
views
Finding a closed form for $ \int_0^1 \frac1x \ln\left(\frac{\ln\left(\frac{1-x}{2}\right)}{\ln\left(\frac{x+1}{2}\right)}\right)\, \mathrm{d}x $
I want a closed form for the following integral
$$
\int_0^1 \frac1x\;\ln\left(\frac {\ln\left(\frac{1-x}{2}\right)}{\ln\left(\frac{x+1}{2}\right)}\right)\, \mathrm{d}x
$$
An integration by parts ...
- 203
2
votes
3
answers
100
views
Solve for $x$: $x^2 + x = 14\log_2(\log_2(x)) + 6, \quad x > 1$
$x^2 + x = 14\log_2(\log_2(x)) + 6, \quad x > 1$
The solutions are $2$ and $4,$ but I couldn't seem to find a way to show it. I've tried substituting and reorganizing terms based on known solutions ...
- 1,426
2
votes
1
answer
79
views
Shouldn't the integral of $1/x$ be $\text{sgn}(x)\ln|x|$?
The integral of $1/x$ across an interval $[-b,-a]$ would be the negative of the integral across the interval $[a,b]$, no? So, the equation ought to be:
$$\int \frac 1x = \text{sgn}(x)\ln|x| + C$$
...
- 1,129
1
vote
2
answers
123
views
How to integrate $\int_{0}^{1} \frac{x \operatorname{Li}_2(1 - x)}{1 + x^2} \, dx$
How to integrate $$\int_{0}^{1} \frac{x \operatorname{Li}_2(1 - x)}{1 + x^2} \, dx$$
My try to integrate
$$\text{I}=\int_{0}^{1} \frac{x \operatorname{Li}_2(1 - x)}{1 + x^2} \, dx$$
\begin{aligned}
&...
5
votes
0
answers
84
views
Can the natural logarithm be meaningfully "natural" in the category theory sense?
This may be more of a pun than anything else, as the two usages of the word "natural" are unrelated, but I was wondering if there was any nontrivial sense in which the natural logarithm ...
- 619