Questions related to real and complex logarithms.

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15
votes
3answers
254 views

Prove that $\int_0^1 \frac{1}{1+\ln^2 x}\,dx = \int_1^\infty \frac{\sin(x-1)}{x}\,dx $

I've found the following identity. $$\int_0^1 \frac{1}{1+\ln^2 x}\,dx = \int_1^\infty \frac{\sin(x-1)}{x}\,dx $$ I could verify it by using CAS, and calculate the integrals in term of ...
1
vote
1answer
99 views

Solve $\log_9 (a) + \log_{12} (b) = \log_{16} (a+b)$ for $a/b$

The question: $$\log_9 (a) + \log_{12} (b) = \log_{16} (a+b)$$ solve for $a/b$. It gives hints: put it all in terms of x. $$9^x=a$$ $$12^x=b$$ $$16^x=a+b$$ Now prove that: $b^2=a(a+b)$ I did and ...
0
votes
1answer
37 views

A question to do with logarithms?

$\log_3x^3 + {3\over \log_3x} =4$ Ok, the way the computer has put it makes it look weird.But it is :log to base 3 of "x" to power 3 plus 3 divided by log to base 3 of x is equals to "4". This ...
3
votes
3answers
59 views

Solving logarithmic equations

The equation that I'm trying to solve is: $$\log _{5x+9}(x^2+6x+9)+\log _{x+3}(5x^2+24x+27)=4$$ Using algebra and principles of logarithms I managed to get the equation down to $$\frac{2\left(\log ...
0
votes
0answers
19 views

Help finding a formula to fit the data - both axes are logarithmic?

I'm an electrical engineer trying to come up with a formula to turn a measurement of light level into lux. The photoresistor changes resistance logarithmically. Lux is also logarithmic. So the data I ...
4
votes
4answers
91 views

Is $\ln(x^{p(x)}) = p(x) \ln(x)$?

I am trying to prove that: $x^{\frac{\ln(\ln(x))}{\ln(x)}} = \ln(x)$ My "solution": $e^{\ln\left(x^{\frac{\ln(\ln(x))}{\ln(x)}}\right)} = e^{\frac{\ln(\ln(x))}{\ln(x)} \ln(x)} = e^{\ln(\ln(x))} = ...
4
votes
1answer
69 views

Is there another function with a property like the log?

Is there another differentiable monotone increasing (or decreasing) function $ f:\mathbb{R} \rightarrow \mathbb{R} $ with a property that $ f(xy) = f(x) + f(y) $, like the log-function has it?
2
votes
6answers
42 views

Solving for $x$ in an exponential equation

Say we the following equation $$F(x) = \frac{\exp(a+bx)}{1 + \exp(a+bx)}$$ Now we set $x=0$ and we want to solve for $a$ as a function of $F_0$. So that, we have: $$F_0 = \frac{\exp(a)}{1 + ...
2
votes
1answer
62 views

Logarithm question, does $\ln \sqrt{7}$ equal to zero?

I got this from my workbook solution, was able to solve the question for the most part but stuck in the last sentence. $$7(\ln\left|x+\sqrt{x^2-7}\right|-\ln\sqrt{7}) + c = 7 ...
1
vote
2answers
44 views

Show using logarithms that the first equation can be transformed into the second.

Show using logarithms that if $y^k = (1-k)zx^k(a)^{-1}$ then $y = (1-k)^{(1/k)}z^{(1/k)}x(a)^{(-1/k)}$.
9
votes
5answers
835 views

How to solve an exponential and logarithmic system of equations?

$$ \left\{\begin{array}{c} e^{2x} + e^y = 800 \\ 3\ln(x) + \ln(y) = 5 \end{array}\right.$$ I understand how to solve system of equations, logarithmic rules, and the fact that $\ln(e^x) = e^{\ln(x)} ...
1
vote
2answers
118 views

Solving $x^{2n} = \frac{1}{2^n}$ for $x$

What is the principle behind solving for a variable that is raised to another variable? I came across this problem doing infinite sums: I had to solve the equation $$x^{2n} = \frac{1}{2^n}$$ for ...
1
vote
3answers
63 views

How to find range of a logarithmic function?

How do I find the range of these logarithmic functions? \begin{align} & \ln(3x^2 -4x +5), \\ & \log_3(5+4x-x^2). \end{align} how should I approach questions like this ? What I did: I found ...
1
vote
3answers
51 views

Proving: $\frac{2x}{2+x}\le \ln (1+x)\le \frac{x}{2}\frac{2+x}{1+x}, \quad x>0.$

$$\begin{equation}\frac{2x}{2+x}\le \ln (1+x)\le \frac{x}{2}\frac{2+x}{1+x}, \quad x>0\end{equation}$$ I found this inequality in this paper: http://ajmaa.org/RGMIA/papers/v7n2/pade.pdf (Equation ...
-1
votes
1answer
36 views

Natural Logarithms (Help) [closed]

Can I have an answer with working out to this question please, I am puzzled at the result I got and don't believe it is right. The question asks to 'Simplify by expressing as a single natural ...
0
votes
0answers
26 views

Manipulating product of two matrices

In a published paper I saw the following $$\log \left(\mathbf{I} + \mathbf{T}\mathbf {Hpp^HH^H}\right)= \log(1+\mathbf {p^HH^HTHp})$$ where uppercase means a matrix while lower case means vector ...
2
votes
1answer
74 views

Is $f(\sqrt{x}) = \sqrt{f(x)}$ true, where $f(xy) = f(x) + f(y)$ for all positives $x, y \in \mathbb{R}$?

Is $f(\sqrt{x}) = \sqrt{f(x)}$ true, where $f(xy) = f(x) + f(y)$ for all positives $x, y \in \mathbb{R}$? It's obviously false. But the point is that "can it be proved without using the fact that the ...
0
votes
1answer
46 views

When does function $(\log_b(x))^p$ change its curvature?

Consider $(\log_b(x))^p$ where $b$ is a constant $>1$; $x, p \in \mathbb R_+$. As we increase the value of $p$ (starting from 1), at specific value of $p$, the curve changes its shape from ...
1
vote
1answer
87 views

Taking Log to find MAXIMIZE summation of variables

I have been reading IEEE papers on communication and in several papers the authors formed objective function like: $\text{Maximize } \sum_k \log r_k $ to maximize the total rate of the system of ...
0
votes
0answers
23 views

What's the non-log form of this equation?

I found this equation in a book and I'd like to know what it would look like in 'non-log' form: $(a \log x + b \log N)^2 + 2g \log x + 2f \log N + C = 0$ Thanks.
1
vote
4answers
66 views

Integrating for a solution in terms of an natural logarithm

Evaluating the following integral: $$\int_1^2 \frac2{1-3x}\ dx$$ why do you have to take the factor of $-2/3$ out when evaluating the integral?
1
vote
0answers
57 views

solve $a\cdot e^{b\cdot x}+c\cdot ln(x)=0$

Is it possible to find the analytical solution of $a\cdot e^{b\cdot x}+c\cdot ln(x)=0$? Is that a transcendental equation?
2
votes
6answers
156 views

How do I solve for $t$ in this equation?

I know I'm supposed to use $\ln()$ to work it out, but I can't remember how it's done. Can anyone help? The equation is $$ 40e^{-t/5}=20 $$
1
vote
2answers
75 views

Does the matrix logarithm always converge for exponential matrices?

We have defined functions on square matrices $X$: $$e^X := I + X + \dfrac{X^2}{2!}+\dfrac{X^3}{3!}$$ and $$log(X):= (X-I)-\dfrac{(X-I)^2}{2} + \dfrac{(X-I)^3}{3}-...$$ The exponential converges for ...
3
votes
2answers
89 views

Evaluation of $\int_{0}^1 \frac{1}{x} \log^3{(1-x)}dx =-\frac{\pi^4}{15}$ and $\int_{-\pi}^{\pi} \log(2\cos{\frac{x}{2}}) dx =0$

In the following encyclopedia, http://m.encyclopedia-of-equation.webnode.jp/including-integral/ I found the relations below \begin{eqnarray} \int_{0}^1 \frac{1}{x} \log^3{(1-x)}dx ...
0
votes
2answers
71 views

Approximation of the Gamma function

I am having trouble obtaining a lower bound for the following formula: $$ \ln\frac{\Gamma\left(\frac{x}{3}\right)}{\Gamma\left(\frac{x}{4}+1\right)\Gamma\left(\frac{x}{12}+1\right)}. $$ I tried using ...
0
votes
2answers
38 views

Find the value of x if 1,$\log_{9}(3^{1-x}+2) $, $ \log_{3}(4.3^x-1) $ are in an AP.

We have an AP: 1, $ \log_{9}(3^{1-x}+2) $, $ \log_{3}(4.3^x-1) $ We have to find value of x. $$ d = a_{2} - a_{1} $$ $$ d = log^{(3^{1-x}+2)}_9 - 1 $$ $$ d = log^{(3^{1-x}+2)^{\frac{1}{2}}}_3 - ...
0
votes
0answers
28 views

solve implicit equation with lambertw, exponentials, logarithms and first order polynom

I have a very complicated problem to solve. I am almost sure it's impossible to solve but maybe one of you guys has a miracle solution for me. I am modelling the behaviour of a photovoltaic cell and ...
1
vote
2answers
101 views

Is the function $\ln(ax + b)$ increasing/decreasing, concave/convex?

$h(x) = \ln(ax + b)$ NB. Examine your results acccording to values of $(a,b)$ I've differentiated twice in order to get the following: $$ h''(x) = -\frac{a^2}{(ax+b)^2} $$ I think this proves ...
0
votes
5answers
115 views

How to resolve $n>(1+\frac{1}{n})^n$?

I'm trying to prove that $\forall n\geq 3, n^{n+1}>(n+1)^n$. I came that this is true for $n>(1+\frac{1}{n})^n$. WolphramAlpha gives $n>2.293166...$ but I failed to compute it analytically.
0
votes
1answer
71 views

Finding value of (y) of logarithmic equation given (x)

I have an logarithmic equation $$\left[ r=a\,e^{b\,\theta} \right] $$ And I plot it to visualise it (see plot below). I can tell by the plot when (t=0), x1=0, y=1 (point AA) but how can I find out ...
0
votes
1answer
54 views

Expotential Growth/Decay - Problem Deriving Atmospheric Pressure Formula

I have a problem deriving the following formula: $$\frac{dP}{dh} = k\left(\frac{P}{T}\right)$$ Using the following 'rule': If $\ \dfrac{dA}{dt} = kA\,$ then $\,A = A_0\left(e^{\,kt}\right)\,$ ...
10
votes
1answer
97 views

Asymptotic behaviour of log log sum

I am interested in the asymptotics of $$F(m) := \prod_{j=1}^m \log(j+1) = \exp\left(\sum_{j=1}^m \log \log(j+1) \right)$$ Is there anything known? If not I figure I will need some good bounds on the ...
1
vote
2answers
100 views

Find the roots of a function with logarithms (possibly using lambert W function)

I am wondering if anyone can help me find an analytical solution to the roots of the following function: $$f(b) = c\log \left( \frac{b}{a} \right) + (n-c)\log \left( \frac{1-b}{1-a} \right),$$ $a,b ...
3
votes
1answer
130 views

Proof of Ramanujan's identity

I'm having trouble understanding Ramanujan's formula from his proof of Bertrand's postulate, namely: $$ \ln \lfloor x\rfloor!=\sum_{k=1}^{\infty}\psi\left(\frac{x}{k}\right) $$ where $ \ln x = ...
1
vote
2answers
59 views

how to solve this equation using logarithm, if not possible how to solve it?

how to solve the following equation: $$0.2948(1-(1+x)^{-5})=x$$ I know to satisfy this equation $x$ should be equal to 0.145 but how i can get there please help!
0
votes
3answers
63 views

Is $\log(-1)$ equal to $-\log(-1)$ [duplicate]

I thought it should be because if the logarithmic identities hold then, $$-\log(-1)=\log(-1^{-1})=\log(-1)$$ But $\log(-1)=i*\pi$ and $-\log(-1)=-i*\pi$
-2
votes
2answers
38 views

Can anyone simplify this logarithmic function? [closed]

The function is: $$X^{\log_a(Y)}$$ Thanks edit: This is from my book. The answer given is y* log (base a) X. But i cant solve. + I am from mobile and never used this site before sorry for ...
2
votes
0answers
179 views

Comparing Large Exponents with different bases.

How to compare large exponents with different bases? Is there any way to roughly approximate their values? For example, sort the elements of list below based on their magnitude. ...
0
votes
1answer
45 views

Very strange logarithm simplification

I have very strange logarithm simplification: $$\begin{array}{l}\frac{{{{\log }_{{2^{{{(x + 1)}^2}}} - 1}}({{\log }_{2{x^2} + 2x + 3}}({x^2} - 2x))}}{{{{\log }_{{2^{{{(x + 1)}^2}}} - 1}}({x^2} + 6x + ...
-3
votes
5answers
51 views

$ \log_{k}x \times \log_{5}k = \log_{x}5 $ find value of $x$. [closed]

If $ \log_{k}x \times \log_{5}k = \log_{x}5 $ ($k$ cannot be equal to $1$ and has to be more than $0$), then how do you find the value of $x$?
24
votes
6answers
3k views

The logarithm is non-linear! Or isn't it?

The logarithm is non-linear Almost unexceptionally, I hear people say that the logarithm was a non-linear function. If asked to prove this, they often do simething like this: $$\ln(x + y) \neq ...
5
votes
2answers
228 views

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

How can we evaluate this definite integral $$I=\int_0^{1/\phi}x\log(x)\log(1+x)\log(1-x)\,dx,$$ where $\displaystyle\phi=\frac{1+\sqrt5}2$ is the golden ratio?
3
votes
4answers
64 views

Using the $\ln(\cdot)$ for $(1-e^{-x})$

The given function: $$B= A(1-(e^{-x}))$$ Now, I want to 'destroy' the e-function by taking the logarithm of it. First, since $\ln(ab) = \ln(a) + \ln(b)$ we get that $\ln(b) = \ln(a) + ...
23
votes
2answers
2k views

Can a logarithm have a function as a base?

For example is $\log_{\sin(x)}(3x)$ a ridiculous equation? I couldn't find an example on any page about logarithms that used a function on a base, but it seems that for an equation like ...
4
votes
2answers
231 views

Ambiguity with Logs and Inequalities

Solve the following inequality: $$0.8^x > 0.4$$ Method 1 (Using the Common Logarithm) $$\log_{10}0.8^x > \log_{10}0.4$$ $$x\log_{10}0.8 > \log_{10}0.4$$ Because $$\log_{10}0.8 < 0$$ ...
0
votes
1answer
21 views

Predictability of what $\lfloor n\log n\rfloor$ “skips”

TL;DR: is there any way to tell what numbers will not be present in the function $\lfloor n\log n\rfloor$ under a given upperbound? I am writing a program that will calculate the sum of the gaps in ...
1
vote
4answers
54 views

How would I solve: $\log_{16} 32 = x$?

How would I solve: $\log_{16} 32 = x$? What I know: 16 is the base 32 is the exponent $$ 32 = 16^x $$ I'm stuck at this point$\ldots$
2
votes
2answers
65 views

Growth Rate $n\ln n$

I mistakenly posted this on MathOverflow. I hope this is a better place for it. I have been investigating a problem about sports teams and came across the function $n\ln(n)$. I want to see if I can ...
8
votes
1answer
194 views

Evaluating $\sum_{n \geq 1}\ln \!\left(1+\frac1{2n}\right) \!\ln\!\left(1+\frac1{2n+1}\right)$

Is there a direct way to evaluate the following series? $$ \sum_{n=1}^{\infty}\ln \!\left(1+\frac1{2n}\right) \!\ln\!\left(1+\frac1{2n+1}\right)=\frac12\ln^2 2. \tag1 $$ I've tried telescoping ...