Questions tagged [logarithms]
Questions related to real and complex logarithms.
10,253
questions
1
vote
0
answers
42
views
What does "any polynomial dominates any logarithm" mean here?
My textbook states that
any polynomial dominates any logarithm: $n$ dominates $(\log n)^3$. This also means, for example, that $n^2$ dominates $n\log n$
However, it wasn't clear to me what the ...
- 2,979
-4
votes
0
answers
38
views
Logarithms find the solution [closed]
What conditions must $a$ and $b$ satisfy for the equation to have at least one real solution? Find all the solutions of this equation:
$1+\log_b(2\log(a)-x)\log_x(b)=2\log_b(x)$
I have tried ...
-1
votes
1
answer
97
views
Solve the power equation $\;2\!\cdot\!3^{x^2}=6^x$
Solve the power equation $\;2\!\cdot\!3^{x^2}=6^x.$
$$2 \cdot 3^{x^{2}} = 6^x $$
$$log_3(9) \cdot log_3(3^{x^{2}}) = log_3(6^x) $$
$$2x^2 - xlog_3(6) = 0 $$
$$x(2x - log_3(6)) = 0$$
$$x = 0$$ or $$...
- 29
2
votes
1
answer
109
views
+50
Generalizing a logarithmic inequality
Let $\{x_i\}_{i=1}^N$ and $\{y_i\}_{i=1}^N$ be real numbers in the interval (0,1). Define for each $i$
$$\alpha_i = x_i (1-x_i) \log^2 \frac{x_i (1-y_i)}{y_i (1-x_i)}$$
and
$$\beta_i = x_i \log \frac{...
- 388
2
votes
1
answer
50
views
Prove $\ln (1+x) \leq x - x^2/4 $ for $x \leq 1$ using Taylor's theorem
RTP: $\ln (1+x) \leq x - \frac{1}{4} x^2$ for $x \leq 1$
I am trying to prove this specifically using Taylor theorem. Here is what I have so far:
$\ln (1+x) = x - \frac{x^2}{2} + \frac{x^3}{3(1+\xi)^3}...
- 487
2
votes
2
answers
183
views
+100
$\operatorname{Li}_{2} \left(\frac{1}{e^{\pi}} \right)$ as a limit of a sum
Working on the same lines as
This/This and
This
I got the following expression for the Dilogarithm $\operatorname{Li}_{2} \left(\frac{1}{e^{\pi}} \right)$:
$$\operatorname{Li}_{2} \left(\frac{1}{e^{\...
- 814
-3
votes
0
answers
25
views
Logarithm and its positive mantissa [closed]
I always get confused while trying to solve logarithm and anti-logarithm values, especially when I have to change the negative values. When do I change the mantissa into negative or keep the negative?
- 1
3
votes
2
answers
275
views
A problem that could use substitution or logs, not sure which works better
This is one of those brain teaser problems on instagram, and it starts here:
$$x^{x^2-2x+1} = 2x + 1$$
And we want to solve for x. My first instinct was to try this
$$\ln(x^{x^2-2x+1}) = \ln(2x + 1)\\
...
- 2,692
3
votes
3
answers
101
views
Why to use modulus in integration of $1/x$ [closed]
$$ \int \frac1x = \log_e |x|+C$$
Why is modulus sign needed. If this is because the domain of logarithmic function is $(0,\infty)$ Then why don't we mention the limitations of the domains of other ...
- 177
-2
votes
0
answers
139
views
Solving $\sqrt{x+1}=-2$ and looking for complex solutions [duplicate]
So the question was
$$\sqrt{x+1}=-2$$
And obviously there is no value for it,
However,
If you do the thing with $e$ and $\ln{}$
$$e^{\ln{\sqrt{x+1}}}$$
and
$$e^{\frac{1}{2}\cdot (\ln{x+1})}$$
Then ...
- 19
1
vote
1
answer
19
views
Rounding to nearest integer in the log domain
This is about the threshold of whether to round down to $N$ or round up to $N+1$ where the proportional rounding error (or the error in the log domain) is smallest.
What is the simplest way to prove
$$...
1
vote
1
answer
77
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 ...
- 865
2
votes
0
answers
22
views
Find the point of intersection of an annuity and an investment
We were completing some classwork for Annuities and finding the Sum of a G.P for simple situations as well as finding the time take for an annuity to reach a given value.
Given the context we wanted ...
0
votes
1
answer
40
views
How to solve for a value in a log
I have a formula:
Weight=onerepmax*(0.488 + 0.538 * ln(-0.075*reps))
And I need to solve for reps given a onerepmax and a weight.
I got as far as:
...
- 113
0
votes
1
answer
46
views
Determining the witnesses (constants $C_0$ and $k_0$) when showing $(log_b n)^c$ is $O(n^d)$ (b > 1 and c,d are positive)
I'm having a hard time finding the constants/witnesses $C_0$ and $k_0$ that show $(\log_b n)^c$ is $O(n^d)$. That is $|(\log_b n)^c| \leq C_0|n^d|$ for $n > k_0$ (b > 1, and c,d are positive).
I ...
- 865