Questions related to real and complex logarithms.

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-2
votes
1answer
45 views

Solve the equation below [on hold]

Solve the equation $$\tan(\cos^{-1}\sqrt{x})=2^{\log_{4}x}.$$ I have no idea where I have to start; it's a little hard for me. So any help?
0
votes
1answer
30 views

LN word problem

measurement of a child's ability to learn is given by the function $$L(t)=\frac{ln(t+1)}{t+1}$$ where t is the child's age in years, for $0 ≤ t ≤ 5$ At what age does a child have the greatest ...
1
vote
2answers
42 views

( Logarithmic Equation ) Solve for x.

$(x+1)^{log(x+1)} = 100(x+1)$ Attempt at solution : $$ (x+1)^{log(x+1)} = 100(x+1)$$ $$= x^{log(x+1)} + 1 = 100x +1$$ $$=(x+1)+1=100x+1$$ $$=−98=99x$$ $$x=−98/99$$ But the answer given in the ...
1
vote
1answer
18 views

Logarithmically bounded function fulfills $f(n) \le \lceil m \cdot \log_b r \rceil$ for certain numbers $n,m,r$

Let $f : \mathbb N \to \mathbb N$ be a function such that $f(n) \le 1 + \log_b n$ for some base $b$ and all $n$. Now let $n \in \mathbb N$ have the property that $$ \frac{r^m - 1}{r-1} \le n < ...
2
votes
2answers
573 views

How to recognise intuitively which functions grow faster asymptotically?

There are some cases where it is not so simple to decide which function grows faster asymptotically. For example, in the following cases, why (intuitively) $g(n)$ should grow faster than $f(n)$, or ...
0
votes
0answers
16 views

What would be the equation for “3% in 100 Hz range, 0.5% in the 2000 Hz range”

The smallest distinguishable pitch/frequency by a human ear is something like this: Pitch is our perceptual interpretation of frequency. As mentioned, ideal human hearing ranges from 20 to 20,000 ...
1
vote
2answers
64 views

Why does $\lim_{x\to0^{-}} \mathrm {Im}\left( \mathrm \ln \left(x\right)e^x\right)=\pi$?

Why does $$\lim_{x\to 0^{-}} \mathrm {Im} \left( \ln\left(x\right) e^x\right)=\pi$$ Obviously this is no coincidence. I was thinking maybe this has to do with Euler's formula, but I don't see how the ...
4
votes
2answers
59 views

Iterative calculation of $\log x$

Suppose one is given an initial approximation of $\log x$, $y_0$, so that: $$y_0 = \log x + \epsilon \approx \log x$$ Here, all that is known about $x$ is that $x>1$. Is there a general method of ...
3
votes
4answers
87 views

How to solve $\lim_{x\to1}\left[\frac{x}{x-1}-\frac{1}{\ln(x)}\right]$ without using L'Hospital's rule?

$$\lim_{x\to1}\left[\frac{x}{x-1}-\frac{1}{\ln(x)}\right]$$ How can I solve this without using the L'Hopital's rule? Any tips or hints would be greatly appreciated. I tried using the substitution ...
-1
votes
2answers
28 views

What is the Solution? x ≥8lgx

x ≥ 8lgx I have to find which x satisfy this inequality. I found the points using graph, but I'd like someone to show me how to find it without it.
3
votes
1answer
137 views

Find the number $n^{2}$ from the number $n^{n^{n^{2}}}$

Find the number $n^{2}$ from the number $\large n^{n^{n^{2}}}$ Any help? I tried with $\log$ but I got nothing.
1
vote
2answers
42 views

Show $n \cdot \log_b r + \log_b \frac{r}{r-1} \le \lceil n \cdot \log_b r \rceil$

Let $n$ be a natural number and $b, r > 1$ be two natural numbers, then I guess we have $$ n \cdot \log_b r + \log_b \frac{r}{r-1} \le \lceil n \cdot \log_b r \rceil. $$ where $\lceil x \rceil = ...
1
vote
1answer
42 views

Logarithm Propr

I'm having a bit of trouble proving the following property: Theorem If $Re(z)>0 $ and $ Re(w)≥0$, then $\log(zw)=\log(z)+\log(w)$, where log is the principal branch. I know that $\log (zw) = ...
3
votes
3answers
193 views

Indefinite integration: $\int x^{x^2+1}(2\ln x+1)dx$

Find the value of the integral: $$\int x^{x^2+1}(2\ln x+1)dx.$$ My attempt: I tried by using integration by parts, but not working since $x^{x^2+1}$ keeps coming again and again. Then I tried putting ...
0
votes
1answer
23 views

Linear, Squared and Logarithmic scales with given input domain and output range

The input domain is $[12,24]$ and the output range is $[0,720]$. Now I know that with using linear scaling the value $16$ of the input range is mapped to $240$; with using sqrt scaling the same value ...
7
votes
4answers
692 views

Limit to infinity with natural logarithms

I found the following problem in my calculus book: Solve: $$\lim_{x\to \infty } \left(\frac{\ln (2 x)}{\ln (x)}\right)^{\ln (x)} $$ I tried to solve it using log rules and l'Hôpital's rule with no ...
0
votes
1answer
22 views

Minimum of the function $b\log_b x$

Why the function $b\log_b x$ has its minimum at $b=e$? How to explain this? I'm asking because I can't understand why ternary base has more economy than binary: ...
0
votes
2answers
36 views

Logarithms and Taylor Series

Before Log Tables, how were they able to compute expressions such as $2^{2.221}$? I understand they could take a Taylor expansion of $\frac{1}{x}$, but how were they able to condense the expansion ...
3
votes
4answers
76 views

Solve for $x$ - Logarithm Equation $\ln x+\ln(x+1)=\ln 2$

My attempt: $\ln x(x+1)=\ln 2$ $e^{\ln x(x+1)}=e^{\ln 2}$ $x(x+1)=2$ $x^2+x-2=0$ $(x-1)(x+2)=0$ therefore $x=1, -2$
0
votes
1answer
16 views

Geometric distribution with given probability value.

The probability of a man hitting a target is $2/3$. If he doesn't stop shooting until he hits the target for the first time, a) What is the probability of taking 5 shots to hit the target? b) Which is ...
-3
votes
1answer
35 views

Express as a single logarithm [closed]

Hi I need to express the following and have no clue how to do so. $$\ln(x+3)-3\ln(x-7)-\ln(x+8)$$ Can someone please help
1
vote
5answers
77 views

logarithm proof fallacious or not?

$e^{-x}=e^{1/x}$ Taking the natural logarithm of both sides $$\ln(e^{-x})=\ln(e^{1/x})$$ $$-x=1/x$$ $$-x^2=1$$ $$x^2=-1$$ $$x=i$$ I know I am doing something wrong here. Also can someone please ...
5
votes
3answers
40 views

Logs - Simplifying with arbitrary constant

I've tried simplifying my answer, which has a constant in it. I would like to know if I am on the right track: $$ \ln(y) = -{x^2\over 2y^2} + C $$ C can be considered as an Arbitrary Constant. From ...
2
votes
5answers
105 views

how to prove that $\ln(1+x)< x$

I want to prove that: $\ln(x+1)< x$. My idea is to define: $f(x) = \ln(x+1) - x$, so: $f'(x) = \dfrac1{1+x} - 1 = \dfrac{-x}{1+x} < 0, \text{ for }x >0$. Which leads to $f(x)<f(0)$, ...
15
votes
0answers
200 views
+200

A difficult logarithmic integral ${\Large\int}_0^1\log(x)\,\log(2+x)\,\log(1+x)\,\log\left(1+x^{-1}\right)dx$

A friend of mine shared this problem with me. As he was told, this integral can be evaluated in a closed form (the result may involve polylogarithms). Despite all our efforts, so far we have not ...
1
vote
1answer
31 views

Imprecise logarithms that reference sets of numbers.

I apologize in advance if my question seems vague, I'm only in algebra II, so It may turn out that I lack the terminology to phrase my question correctly. Some background, we just finished our unit ...
1
vote
0answers
49 views

Are logarithms of prime numbers algebraically independent?

From Baker's theorem it follows that a linear combination of natural logarithms of prime numbers with non-zero algebraic coefficients can never be zero. Has it been proved that the set of all natural ...
1
vote
2answers
49 views

Summation of a logarithmic series for $\ln(2(r^2 - 1)/r^2)$

Given that $$\sum_{r=2}^{n}\ln\frac{r^2-1}{r^2}=\ln\frac{n+1}{2n}$$ for $n >1$. Express $$\sum_{r=32}^{62}{\ln\frac{2(r^2-1)}{r^2}}$$ as $$A\ln 2 + B\ln3 + C\ln7$$ where $A$, $B$, $C$ are positive ...
-3
votes
0answers
37 views

logarithmic function? [closed]

Zeus Industries bought a computer for $2149. It is expected to depreciate at a rate of 25% per year. What will the value of the computer be in 3 years? Round to the nearest penny. Do not type the ...
0
votes
1answer
36 views

How to solve integral with natural logarithm and product

I am trying to solve the following integral: $$\int{\frac{x}{4} \ln\left(\frac{4}{x}\right)}$$ Using this integral table, the more close case is (43). However, this is not the right one to use. Do ...
3
votes
1answer
68 views
+50

Checking logarithm inequality.

Which one of the following is true. $(a.)\ \log_{17} 298=\log_{19} 375 \quad \quad \quad \quad (b.)\ \log_{17} 298<\log_{19} 375\\ (c.)\ \log_{17} 298>\log_{19} 375 \quad \quad ...
2
votes
1answer
37 views

Why does this sequence of operations give $x^{\frac{1}{x-1}}$?

I found (purely from experimentation) that if you start with a random number and successively: Exponentiate, Raise to the power of $x$, Take the log with the same base as step one, Take the $x$-th ...
9
votes
0answers
93 views

A conjectured identity for tetralogarithms $\operatorname{Li}_4$

I experimentally discovered (using PSLQ) the following conjectured tetralogarithm identity: $$\begin{align}&\phantom{+\;}19\!\;\pi^4-570\ln^42-90\ln^43\\ ...
4
votes
2answers
50 views

Why are values greater than $\pi$ radians given as negative in exponential form?

Find the fifth roots of $-3+3i$ in exponential form. My answers are: $$1.335e^{3i\pi/20}$$ $$1.335e^{11i\pi/20}$$ $$1.335e^{19i\pi/20}$$ $$1.335e^{27i\pi/20}$$ $$1.335e^{35i\pi/20}$$ Wolfram ...
11
votes
2answers
166 views

Integral $\int_0^1\frac{\log(x)\log(1+x)}{\sqrt{1-x}}\,dx$

I'm trying to evaluate this definite integral: $$\int_0^1\frac{\log(x) \log(1+x)}{\sqrt{1-x}} dx$$ It's clear that the result can be expressed in terms of derivatives of a hypergeometric function with ...
7
votes
8answers
177 views

Calculate $\ln 97$ and $\log_{10} 97$

Calculate $\ln 97$ and $\log_{10} 97$ without calculator accurate up to $2$ decimal places. I have rote some value of logs of prime numbers up to $11$. $97$ is a little big. In case it would ...
2
votes
0answers
30 views

Tweaking Reddit's Ranking Algorithm

This image explains how Reddit's Ranking algorithm works. As you know, Reddit is a very high traffic site. Therefore, the post rank decreases quite fast. This algorithm puts emphasis on bringing ...
1
vote
1answer
27 views

Multiplication using addition using logarithms

Multiplication by addition using logarithms is possible and took place in past using slide rule and log tables. Is it still used in software? Maybe sometimes it's faster to convert numbers and use ...
3
votes
4answers
63 views

Evaluating the limit: $\lim_{x\rightarrow -1^+}\sqrt[3]{x+1}\ln(x+1)$

I need to solve this question: $$\lim_{x\rightarrow -1^+}\sqrt[3]{x+1}\ln(x+1)$$ I tried the graphical method and observed that the graph was approaching $0$ as $x$ approached $-1$ but I need to know ...
8
votes
3answers
114 views

Logarithmic Integral II

While reviewing an old calculus book the following integral was assigned: \begin{align} \int_{0}^{1} \left( x^{a-1} - x^{n-a-1} \right) \, \frac{\ln^{2}x \, dx}{1-x^{n}} = \frac{2 \, \pi^{3} \, ...
1
vote
2answers
41 views

how to find log base 2 of decimal number without calculator

As with calculator things are simple but I don't know how to calculate log base 2 of decimal number without calculator. like $\log_2(0.25)$ etc.
1
vote
1answer
40 views

Requirement for a given function to be smooth

I have quite a basic question about the derivatives. My uncertainty comes mainly from the fact that I don't know how the complex logarithm behaves. Here is the description (this task is not ...
4
votes
3answers
78 views

Simple Logarithms Equation

$$3^x = 3 - x$$ I have to prove that only one solution exists, and then find that one solution. My approach has been the following: $$\log 3^x = \log (3 - x)$$ $$x\log 3 = \log (3 - x)$$ $$\log 3 ...
1
vote
0answers
31 views

Create log-normal on y axis?

I currently have a graph with log numbers on the x-axis and the y-axis goes from 0-100. How can I get it to I guess log normal y-axis as shown in the picture below? Thank you for your help!
6
votes
5answers
80 views

Solve for $x$ : $\log_3(3x + 2) = \log_9(4x + 5)$

Solve for $x$ $$ \log_3(3x + 2) = \log_9(4x + 5) $$ I changed the bases of the logs $$ \frac {\log_{10}(3x + 2)} {\log_{10}(3)} = \frac {\log_{10}(4x + 5)} {\log_{10}(9)} $$ Now I'm stuck, ...
1
vote
2answers
49 views

Show that $a^{\log_c b}= b^{\log_c a}$

Show that $a^{\log_c b}= b^{\log_c a}$. I start from LHS and add $\log a$ on it, but it leave $\log_c b$. Then I have not idea about how to continue it(maybe my working is wrong... Can anybody solve ...
2
votes
1answer
20 views

Are there more convenient ways of getting the number of digits of a positive integer?

I want to define $n$th power of $10$ for a positive integer. Say for $43$ it would be $2$, for $5$ it would be $1$, for $9999$ it would be $4$. As for $1$, $10$, $100$, ... I am still shifting between ...
5
votes
3answers
542 views

Basic Logarithm Equation

$\log_2(x) = \log_x(2) $ Using the change of base theorem: $\dfrac{\log(x)}{\log(2)} = \dfrac{\log(2)}{\log(x)}$ Multiplied the denominators on both sides: $\log(x)\log(x) = \log(2)\log(2)$ I kind ...
1
vote
7answers
137 views

How do I solve the equation $e^{\ln(2x+1)} = 5x$?

The problem is $$e^{\ln(2x+1)} =5x$$ I've tried using natural logs to both sides like.. $2x+1= \ln 5x $ But I'm not sure if $\ln$ and $e^{\ln}$ cancel out.
0
votes
4answers
59 views

Given $x^2 + y^2 = 34xy$, show that $\log\left(\frac{x+y}{6}\right)= \frac{\log x + \log y}{2}$

If $x^2 + y^2 = 34xy$, show that $$\log\left(\frac{x+y}6\right)= \frac{\log x + \log y}{2}.$$ I tried to put log into the first equation, but I have no idea about how the $34$ being simplified in the ...