0
votes
0answers
15 views

Can't find solution to Calculus 8th (Adams, Essex) problem

I've been sitting here for hours trying to find a solution to his problem. If you have the function $g(y)$, which is the inverse of $f(x) = x^x,\\ e^{-1} \leq x < \infty,$ show that ...
2
votes
0answers
62 views

${\mathfrak{I}} \int_{0}^{\pi/2} \frac{x^2}{x^2+\log ^2(-2\cos x)} \:\mathrm{d}x$ and $\int_{0}^{\pi/2} \frac{\log \cos x}{x^2}\:\mathrm{d}x$

I have found the following new result connecting to rational log-cosine integrals. Proposition. \begin{align} \displaystyle & {\mathfrak{I}} \int_{0}^{\pi/2} \frac{x^2}{x^2+\log ^2(-2\cos x)} ...
0
votes
0answers
40 views

solving the logaritham [duplicate]

I was trying to solve: $\log_2 x \cdot \log_{0.25} x \cdot \log_{0.125} x \cdot \log_{16} x > \frac {2}3$ heres my attempt at it; using logaritham laws and a little algebra we get from $\log_2 x ...
2
votes
2answers
49 views

I need help on the process of solving this derivative.

How do I go about solving this derivative. $$f(x)=\ln\left(\frac{7x}{x+4}\right)$$ I go from this to $$1. \quad f(x)=\ln(7)+\ln(x)-\ln(x+4)$$ and then $$2. \quad f'(x)=\frac{1}{x}-\frac{1}{x+4}$$ then ...
2
votes
1answer
70 views

Checking derivation of y = a^x

Can you tell me if there are any flaws with this derivation of $y = a^x$... The assumptions are that the derivative $$\frac{d}{dx}e^x = e^x$$ and that the derivative $$\frac{d}{dx}\ln x = ...
2
votes
1answer
55 views

What does this log notation mean?

Can someone please explain what $^2\log x$ means? Is it the same as saying $\log x^2$ or is it something completely different? Here is an image of it as an example:
0
votes
2answers
35 views

Logorithms on a first level learning

Solve log$_{5x-1}$ $4$ $=$ $1/3$ $(5x-1)^{1/3}$=4 $((5x-1)^{1/3})^3$ = $4^3$ $5x-1=64$ $5x=65$ $13$ I am not sure where to go with this. I learned some things about logs before my class ended ...
0
votes
3answers
73 views

Question releating to the $\int^x_1\frac{\ln(t)}{t+1}$

If $f(x)=\int^x_1\frac{\ln(t)}{t+1}dt$ if $x > 0$. Compute $f(x) + f(1/x)$. As a check, you should obtain $f(2)+f(1/2)=(\ln2)^2$ I have tried evaluating the integral ...
0
votes
0answers
27 views

Do so big $p \in \Bbb N : \lim_{n \to \infty} \frac{\ln^p {n} }{n} = A \ne 0, A \in \Bbb R$ exist?

We know $$\lim_{n \to \infty} \frac{\ln {n} }{n} = 0$$ $$\lim_{n \to \infty} \frac{\ln^n {n} }{n} = \lim_{n \to \infty} \frac{n\ln^{n-1} {n} }{1} = \infty$$ For usual $p \in \Bbb N $: $$\lim_{n \to ...
2
votes
0answers
107 views

What is ${\mathfrak{R}} \int_{0}^{\pi/2} \frac{x^2}{x^2+\log ^2(-2\cos x)} \:\mathrm{d}x$?

This is a new integral that I propose to evaluate in closed form: $$ {\mathfrak{R}} \int_{0}^{\pi/2} \frac{x^2}{x^2+\log ^2(-2\cos x)} \:\mathrm{d}x$$ where $\log (z)$ denotes the principal value of ...
1
vote
2answers
56 views

Is the inverse ackermann function the slowest growing function that goes to infinity?

Actually, this is not precisely my question. If $a(x)$ is the inverse ackermann function, then obviously $a(a(x))$ grows slower than $a(x)$, as does $\log(a(x))$, and so on. But is there a function f ...
1
vote
1answer
22 views

If $g(x) = \text{arctanh}\ (\log x)$, find $g'(x)$.

If $g(x) = \text{arctanh}\ (\log x)$, find $g'(x)$. I tried to separate the terms first and I got $\dfrac12 (\log(1+\log x) - \log(1-\log x))$. The answer is $\dfrac1{x(1-\log x)^2}$.
1
vote
0answers
54 views

Logarithm and “basic” functions.

To express the antiderivatives of $\frac{1}{x}$, we cannot apply the formula $\int x^n dx=\frac{x^{n+1}}{n+1}+C$ and we need to introduce a new function, the logarithm. But how can we prove that ...
2
votes
1answer
75 views

Tighter logarithmic inequality

There is a well-known lower bound for $$ x\log{1+x\over x}\geq {x\over1+x} $$ for $x\geq0$. I know a tighter lower bound on the same domain $$ x\log{1+x\over x}\geq{2x\over1+2x}\geq {x\over1+x}. $$ It ...
2
votes
1answer
65 views

$\iiint \frac{1}{x^2+y^2+(z-2)^2}dA$ where $A=\{x^2+y^2+z^2 \leq 1\}$ check my answer!

I would like someone to review my solution please, the original question is to calculate $\iiint \frac{1}{x^2+y^2+(z-2)^2}dA$ where $A=\{x^2+y^2+z^2 \leq 1\}$ What I did: First I changed variables ...
0
votes
1answer
63 views

How to simplify $\ln^2\left(x\right)+2 \ln x-3$

I dont know how to simplify $\ln^2\left(x\right)+2 \ln x-3$ I dont know how to get $(\ln(x)+1)(\ln(x)+3)$ But I am stuck and don't really know how to do that. I tried something like this: $2\ln ...
0
votes
1answer
94 views

Unable to comprehend a connection between two equations

I was reading this paper and got stuck at the transition from Equation (13) to Equation (14) (p. 16/17). We got a function of the form: $y(t)=k(t)^{\alpha}h(t)^{\beta}$ We know it grows from zero ...
1
vote
2answers
75 views

How can we relate calculus, trigonometry etc in real life

I have always wondered what does trigonometry, calculus, logarithms solve real world problems? Where do they apply in real life? Is there any simple book where I can understand it?
7
votes
4answers
359 views

Hints on calculating the integral $\int_0^1\frac{x^{19}-1}{\ln x}\,dx$

I would be happy to get some hints on the following integral: $$ \int_0^1\frac{x^{19}-1}{\ln x}\,dx $$
0
votes
1answer
58 views

Finding time constants of a circuit?

So this is a homework question and I am having trouble figuring out what they are asking. 'The potential difference (voltage) across the capacitor at time t > 0 is given by $V_C(t) = q(t)/C$. The ...
5
votes
1answer
144 views

Derivative of $f(x)^{g(x)}$ at points when $f(x)=0$

I am interested in understanding the general behavior of the derivative for $$f(x)^{g(x)}$$ at points where $f(x)=0$. For example, if $f^g=x^n$ we have $$\frac{d}{dx}f^g(0)=\begin{cases}0 & n\ge ...
-2
votes
1answer
33 views

How to solve for $k$ when the area about the $x$ axis and under the graph of the $f(x) = \frac1x$ from interval $x = [2, k]$ is equal to $\ln(4)$?

What approach would be ideal in solving for a number $k$ when the area about the $x$ axis and under the graph of the function $f(x) = \frac1x$ from interval $x = [2, k]$ is equal to $\ln(4)$?
5
votes
3answers
64 views

Why does $\int^{ab}_{a} \frac{1}{x} dx = \int^{b}_{1} \frac{1}{t} dt$?

I can't understand how the integral having limits from $a$ to $ab$ in Step 1 is equivalent to the integral having limits from $1$ to $b$. I'm a beginner here. Please explain in detail. ...
2
votes
4answers
253 views

Derive an equation for derivative of ln x

$\frac{d}{dx}e^x = e^x$ use this fact together with the definition of the natural log $\ln x$ as the inverse of the function of $e^x$ to derive an equation for the derivative of $\ln x$.
1
vote
3answers
120 views

Convergence of series minus logarithm

im trying to solve this problem since two, three days.. Is there someone who can help me to solve this problem step by step. I really want to understand & solve this! $$ Show\ \exists \ \beta ...
0
votes
0answers
25 views

Summation of a function with the variable both in the function amd in the upper limit

E is defined as : E = c1 ( a$\rho$ + b$\rho ^{2}$ ) + c2 $\rho$ ( c + d $\sum_{j=0}^{n} (\log{ \frac{R\rho}{j} } ) $ ) + c3 $\rho ^{2}$ a, b, c, d, c1, c2, c3, R are known constants. $\rho$ is the ...
2
votes
1answer
64 views

Separating the log of a sum

I know there is no formula to separate the log of a sum, e.g. $\log(X+Y)$ into two parts, but are there any approximation rules that can allow me to achieve this objective? ...
0
votes
1answer
33 views

Prove this logarithm equation

I keep getting the wrong answer. Can someone please correct my working out a^x=b^(1-x) In(a)^x=In(b)^(1-x) xIn(a)=(1-x)In(b) xIn(a)=In(e)-xIn(b) xIn(a)+xIn(b)=In(e) x[In(a)+In(b)]=Ine ...
2
votes
2answers
57 views

Limit of a Logarithm with Different Bases

We are to compute $$\lim_{n->\infty}{\frac{2^{\log_3 n}}{3^{\log_2 n}}}$$ Clearly the bases are reversed between the logarithm and exponents, so I can't seem to find any logarithm or exponential ...
0
votes
1answer
44 views

Inverse Function of Logarithm

The answer is A but I don't understand why! $ -2 \log_e (x^2) $ can be re-written as $ -4 \log_e(x) $ right? but why do these two graphs look different? the graph $-2 \log_e (x^2) $ is one to ...
0
votes
1answer
17 views

Maximum value of constant in logarithm problem

The first thing I did was: make: (x-1)^2 - k > 0 (x-1)^2 > k don't know what to do after this point... the maximum value of k is 9 i dont really understand what the maximum value of k is? ...
1
vote
3answers
80 views

Evaluate the integral. $\int x^2 \log(4x) dx$

The problem is $\int x^2 \log(4x) dx$ Here $\ln$ refers to the natural logarithm. So far, I know $u = x^2$ and $du = 2x (dx)$. So $dv = \ln(4x) dx$ and $v = 1/x$, but I don't know where to go from ...
1
vote
0answers
84 views

Showing that a logarithmic inequality holds

Given $0 < x_1 < x_2 < x_3 < x_4 < 1$, how can I show that the following inequality holds: $$ \frac{1}{R(x_1, x_3)}+\frac{1}{R(x_2, x_4)}<\frac{1}{R(x_1, x_2)}+\frac{1}{R(x_3, x_4)} ...
3
votes
2answers
71 views

Exponents with the same power

I've wanted to practice solving simple operations on exponents, so I've made a couple of equations to which I know the answers. $$5^x -4^x = 9$$ I feel really stupid, because I can't solve this one ...
17
votes
2answers
320 views

Integral $\int_0^1\frac{\ln x}{x-1}\ln\left(1+\frac1{\ln^2x}\right)dx$

Is it possible to evaluate this integral in a closed form? $$ I \equiv \int_{0}^{1}{\ln\left(x\right) \over x - 1}\, \ln\left(1 + {1 \over \ln^{2}\left(x\right)}\right)\,{\rm d}x $$ Numerically, ...
1
vote
3answers
34 views

Complex derivative involving exponents and natural log

Find: $\frac{d}{dx} a^{x\ln x}$ I have tried several methods involving u-substitution etc, but can't figure it out.
0
votes
3answers
65 views

$(5^{2x}-1)(5^x)=1/5^x$ solve

I have the problem $(5^{2x}-1)(5^x) = 1/5^x$. I have already simplified it to $5^{3x}-1=1/5^x$ My question is when I do $\log$ base $5$ to the left side of the equation to get $3x-1$ by itself so ...
0
votes
1answer
58 views

Integration involving $\log_2(x)$

Having a hard time going about this problem: $$\int{\frac{\ln(2)\log_2(x)}{x}}$$ I believe $\ln(2)$ would be considered a constant, so than the equation would then changed to: ...
2
votes
0answers
48 views

show that $(x^b+y^b)^{1/b} < (x^a+y^a)^{1/a}$ if $x>0$, $y>0$, and $0<a<b$

Show that $(x^b+y^b)^{1/b} < (x^a+y^a)^{1/a}$ if $x>0$, $y>0$,and $0<a<b$ by examining the sign of the derivative of an appropriate function. This is an exercise in middle part of ...
3
votes
2answers
90 views

Apply the natural logarithm fractional number of times

Let f_n(x) be the recursive function that adds 1 to x and takes the natural logarithm, ...
27
votes
5answers
324 views

An integral with irrational exponents $\int_0^\infty\frac{\log\left(\frac{1+x^{4+\sqrt{15}}}{1+x^{2+\sqrt{3}}}\right)}{\left(1+x^2\right)\log x}dx$

I was challenged to prove this identity $$\int_0^\infty\frac{\log\left(\frac{1+x^{4+\sqrt{15\vphantom{\large A}}}}{1+x^{2+\sqrt{3\vphantom{\large A}}}}\right)}{\left(1+x^2\right)\log ...
4
votes
0answers
83 views

Differentiating $y=x^x$ with the formal definition of a derivative

A friend and I were messing around with derivatives, and while we both know the procedure for finding the derivative of $y=x^x$ with logarithmic differentiation, i.e. $$y=x^x\\ ln(y)=x*ln(x)\\ ...
1
vote
3answers
39 views

How to move from powers to simple logarithms?

I'm following a book that briefly moves from $$16000 \times 2^{\displaystyle \left (-\frac{x}{24} \right )} = 1600$$ to $$x = \frac{24 (\log(2) + \log(5))}{\log(2)}$$ adding the comments that ...
4
votes
6answers
107 views

$\lim_{n\to\infty}\left(1+\frac{3}{n}\right)^\frac{n}{2}$

I am trying to resolve this to number $e$. However, I would like to do it in the simplest form. just a note I already tried wolfram but I would like someone to give me a simpler solution. ...
10
votes
2answers
264 views

$\lfloor n^{1/2}\rfloor+\cdots+\lfloor n^{1/n}\rfloor=\lfloor \log_2n\rfloor +\cdots+\lfloor \log_nn \rfloor$

Prove that: $\lfloor n^{1/2}\rfloor+\cdots+\lfloor n^{1/n}\rfloor=\lfloor \log_2n\rfloor +\cdots+\lfloor \log_nn \rfloor$, for $n > 1,\, n\in \mathbb{N}$ For example. For $n=2$, we have $\lfloor ...
2
votes
2answers
296 views

Why can't I use product rule to derive x ln(3)?

The product rule is defined as $$(f \cdot g)' = f' \cdot g + g' \cdot f.$$ I have the following function $u(x) = x\cdot \ln(3)$. I understand that you can derive it by implicit differentiation and ...
0
votes
3answers
58 views

$\ln( \exp( \ln( \exp( 64 )^{1/2} )^{1/2} )^{1/2} )$

I keep getting the answer 8. But the textbook as well as wolfram say it's 8^(1/2) or in other words 2(2^(1/2)). Here are the steps I took, basically just following the rules of logarithms. ...
1
vote
2answers
38 views

Taking the logarithmic derivative of an exponential difference function after applying L'Hospital's Rule

Can somebody please explain the following application of L’Hospital’s Rule? Find the limit: $$\lim_{x\rightarrow 0} \frac{5^x-3^x}{x}$$ Solution: Determining that this function has indeterminate ...
5
votes
2answers
2k views

U-substitution for integral of 1/(1+e^x)dx. What am I doing wrong?

Here is my work, witth the right answer. I feel like every step is right, but somehow I am getting the wrong answer. How? $$ \int \frac{1}{1+e^z}dz = \int\frac{1}{e^z(\frac{1}{e^z} + 1)}dz ...
0
votes
1answer
173 views

Logarithmic inequalities

Full disclosure: This is a homework problem, but my question is regarding a concept that came about during solving the problem, not the actual solution to the problem. Problem: Rewrite as geometric ...