Questions related to real and complex logarithms.

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1
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5answers
63 views

Rearrange the equation and solve for $y(x)$ [closed]

How do I solve for $y$? $$ \ln\left(\frac{y-1}{y+1}\right)= x^2 $$
2
votes
2answers
55 views

Is it possible to use complex logarithm to integrate $1/(z+i)$ along a path?

Evaluate the following on the path $\gamma_1$ with endpoints $[-1,1+i]$ $$ \begin{align} I_1=\frac{i}{2}\int_{\gamma_1} \frac{1}{z+i}dz -\frac{i}{2}\int_{\gamma_1}\frac{1}{z-i}dz \end{align} $$ Am I ...
0
votes
2answers
57 views

What is the method to correctly isolate $y$ as the dependent variable for $x = e^y$?

In this youtube video about 5:00 minutes in, the instructor makes the point that you can simply exchange the $x$ and $y$ values of the exponential form $x = e^y$ of the equation $y = ln x$ to make $y$ ...
9
votes
2answers
330 views

A series with only rational terms for $\ln \ln 2$

We all know that $$ \ln 2 = \sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{n}. $$ Do you know a series with only rational terms for $$\ln \ln 2 = ?$$ Let's exclude base expansions with non ...
0
votes
1answer
113 views

Finding the transfinite diameter of the level sets of complex logarithm

Given a simply-connected domain $|g(z)|\ge C$ how can I find the analytic conformal mapping guaranteed by the Riemann mapping theorem? In particular I'm interested in finding the transfinite diameter ...
2
votes
2answers
39 views

Expansion of Logarithms with Cube Roots

Does the following expand to the following $$ \log_6(11^6\sqrt[3]{12}) $$ = $ 6\log_6(11) + \log_6 (\sqrt[3]{12})$
6
votes
3answers
184 views

Logarithm Equality

$$\sqrt{\log_x\left(\sqrt{3x}\right)} \cdot \log_3 x = -1$$ I am not entirely sure how to go about solving for $x$. I cannot square each side because the product isn't $≥ 0$, I can't think of any ...
0
votes
0answers
47 views

Is this double limit for logarithms true?

Mathematica knows that: $$\gamma = \lim_{n\to \infty } \, \lim_{s\to 0} \, \left(\int \frac{(s+1)^{-\exp (n)-1}+s-1}{s} \, ds+\frac{(s+1)^{-n-1}+s-1}{s}\right)$$ Where $\gamma$ is Euler Gamma ...
0
votes
2answers
56 views

Show that $\log[(1+i)^2]\neq 2\log(1+i)$

The problem is as stated in the title. I found that the $\mathrm{Log}[(1+i)^2] = 2\mathrm{Log}(1+i)$. We know that $$\mathrm{Log}(z)=\ln(r)+i\theta$$ Now, without defining a branch, doesn't that mean ...
2
votes
1answer
56 views

Proof of the analyticity of complex logarithm

Let $a\in(-\pi,\pi]$ and $f:G\to\mathbb C$, $G = \{ z\in\mathbb C\setminus\{0\},\operatorname{Arg}z\neq a \}$ $$f(z)=\ln|z|+\imath \arg_a z,\quad a<\arg_az<a+2\pi$$ Prove that $f$ is ...
0
votes
2answers
37 views

Solving equations having both log and exponential forms

How can one Solve equations having both log and exponential forms: For eg... $e^x$ $=$ $\log_{0.001}(x)$ gives $x=0.000993$ (according to wolfram-alpha ...
-1
votes
2answers
62 views

Is $f(x) = 2 + \ln x$ another way to write $f(x) =\log_e x +2$?

I just want to make sure I am correctly understaning this concept. $f(x) = 2 + \ln x$ is the same as $f(x) =\log_e x +2$ Thus my T graph would look like so: e^y|x+2 -3|2.049 -2|2.135 ...
0
votes
3answers
28 views

How do I rewrite a logarithm in exponential form, so as to plot it? $f(x) = 2\log x$

How do I write $f(x)=2\log x$ in exponential form? Is $2(10)^y=x$ correct?
2
votes
2answers
54 views

Does loga/logb = log(a^(1/logb))?

I know $\log(a^b)=b\log(a)$. However, Wolfram Alpha tells me that $\frac{\log(a)}{\log(b)}$ does not equal $\log(a^\frac{1}{\log(b)})$. Is Wolfram Alpha correct? If it is, why is it correct? I'm ...
0
votes
1answer
15 views

What is the logaritmic form of $v=Ae^{Bi}$

I am reading a scientific paper, which uses a model of the form $v=Ae^{Bi}$ and then it says that this model has the following logarithmic form $\ln (v) = Bi + ln(A)$ where A is a constant. But the ...
0
votes
2answers
37 views

How to use the Comparison Test to investigate the convergence of $\sum (\ln n)/n^\alpha$?

Let $$\sum\limits_{n=1}^\infty \frac{\ln n}{n^\alpha}, \alpha\in\Bbb{R}$$ I need to investigate the convergence of this series. I've read that since the series is positive for all $n$ then it ...
0
votes
1answer
12 views

Multivariable-calculus, logarithms

I got the function $f(x,y)=\ln(1+x^2+y^2)$. There are three tasks to answer. a)Decide the function´s stationary points and classify them if possible. Here I got the answer to $(0,0)$ is a local ...
7
votes
1answer
96 views

How to solve $\log _{x^{2}-3}(x^{2}+6x)<\log _{x}(x+2)$?

How to solve the following inequality $$\log _{x^{2}-3}\left(x^{2}+6x\right)<\log _{x}(x+2)\ ?$$
0
votes
2answers
53 views

A Method For Calculating Large Exponents Quickly

I've derived a formula for calculating large exponents quickly: $$a^b = 2 \cosh( - b \log( a ) )$$ My question is: Has anyone seen anything similar? I am curious if either it's novel OR if I have ...
2
votes
5answers
203 views

Can this log question be simplified?

$ { 2^{log_3 5}} -  {5^{log_3 2}}.$ I don't know any formula that can apply to it or is there a formula?? Even a hint will be helpful.
1
vote
1answer
44 views

Find real-valued sequences $x(n)$ for which $c^{x(n)} = o(1/n )$

For which $x=x(n)$ does it hold that $$c^x = o\left(\frac{1}{n}\right)$$ where $c\in(0,1)$ is a constant. So clearly, for $x=n$, this is true. But for which $x =o(n)$ does this hold? I thought ...
0
votes
1answer
28 views

How to scale a equation e.g. by log

I'm currently trying to scale an equation since the numbers I have to calculate with are pretty large and Matlab outputs Infinity (Inf). However, the question here is more about the mathematics behind ...
4
votes
1answer
27 views

Generalize logarithmic coincidences

After playing around with logarithms, I've found the following coincidences: $\log_{10}{2} \approx 0.3$, since $2^{10} \approx 10^3$, and $\log_{10}{5} \approx 0.7$, since $5^{1000} \approx ...
0
votes
2answers
145 views

Find all real solutions to the equation $3^{1 + 2\log_3(y-x)} = 48$

$$3^{1 + 2\log_3(y-x)} = 48$$ With this problem I have difficulty getting rid of the exponent. $2\log_5(2y - x - 12) = \log_5(y-x) + \log_5(y + x)$
5
votes
4answers
195 views

To find the logarithm of $1728$ to the base $2 \sqrt{3}$

Find the logarithm of: $1728$ to base $2\sqrt{3}$. Let, $\log_{2\sqrt{3}} 1728 = y$, then $$\begin{align} (2\sqrt{3})^y &= 1728\\ 2^y(\sqrt3)^y &= 1728\\2^y(3^\frac12)^y &= ...
1
vote
2answers
933 views

The logarithm of 3 base 10 is irrational

Prove that the logarithm of 3 base 10 is irrational The Fundamental Theorem of Arithmetic is that every integer is a product of primes. So far I have, Suppose $\log_{10}(5)$ is rational. Then ...
0
votes
2answers
20 views

Taking the logarithm of both sides of $x=(ab/b)^y$

I got confused at $x=(\frac{ab}{b})^y$. If you take the logarithm with base $b$ of $x=(\frac{ab}{b})^y$, wouldn't it be $y\log{bx}=\frac{ab}{b}$. Why did the solution suddenly take the log of ...
2
votes
1answer
68 views

How to solve $6^{2x}-10\cdot 6^x=-21$ using logarithms?

What do I do with $\large 6^{2x}-10\cdot 6^x=-21$? Since $6$ and $-60$ are not of the same base (nor can they be written as exponents of the same base cleanly) I am having trouble solving for ...
2
votes
4answers
67 views

How to solve this kind of equation $(x^y=y^x)$

I'm little bit stuck with this system of equations : $x^y=y^x$ and $x^3=y^2$ An obvious solution is $(x,y) = (1,1)$ but what about the solution $(9/4,27/8)$ ? I know the relation $a^r=e^{r ...
1
vote
3answers
62 views

How to use the logarithm method to solve $18^{4x-3}=(54\sqrt{2})^{3x-4}$ for $x$?

What value will satisfy this equation: $$18^{4x-3}=(54\sqrt{2})^{3x-4}$$ Please use the logarithm method. I am having a problem in expressing $54\sqrt{2}$ in the power of $18$. My book simply ...
0
votes
2answers
45 views

Dealing with Logarithms. $\log(b^x + a) = \log(c)$

What methods/techniques are available to solve for x in the following type of situation: $$ \log(b ^x+a)=\log( c ) $$ The only log methods I have been exposed to are using the power laws and bring x ...
2
votes
4answers
80 views

Exponential equation: $2e^{-x} - e^{-2x}=0.$ [closed]

$2e^{-x} - e^{-2x}=0.$ the correct answer is $x=-\ln2$. How do I get there?
0
votes
0answers
16 views

Normalizing Data for Graph

Firstly, sorry for the long post, but I must be detailed in my explanation here. This is a computer science heavy topic, and I've posted it on the CS section of Stack Overflow already, but the main ...
4
votes
3answers
119 views

Solve for $x$ in the equation [closed]

Please help me to solve for x using maybe logarithm or exponential rules (or both) $$ 5^x=2 \cdot 3^x $$
66
votes
16answers
10k views

How do you explain the concept of logarithm to a five year old?

Okay I understand that it cannot be explained to a 5 year old. But, how do you explain the logarithm to primary school students?
11
votes
3answers
623 views

What is the $x$ in $\log_b x$ called?

In $b^a = x$, $b$ is the base, a is the exponent and $x$ is the result of the operation. But in its logarithm counterpart, $\log_{b}(x) = a$, $b$ is still the base, and $a$ is now the result. What is ...
0
votes
3answers
92 views

Solution to $x=100e^{-x/100}$?

How do you solve $$x=100e^{-x/100}$$ If I use $\ln$ then $\ln(x/100)=-x/100$. How do I get from that to $x=56.7$?
0
votes
3answers
40 views

Removing logs from equation

I have a simple question that I need clarification on: If $$\log(a) = \log(b) + c$$ is it true that $$a = b + \exp(c)$$ Is this correct or am I missing something really basic that I cant ...
3
votes
3answers
127 views

$\int_{0}^{\pi/2}\ln\left(1+4\sin^4 x\right)\mathrm{d}x$ and the golden ratio

We already know that, for any real number $t$ such that $t\geq-1$, $$ \int_{0}^{\pi/2} \ln \left(1+t \sin^2 x\right) \mathrm{d}x = \pi \ln \left( \frac{1+\sqrt{1+t}}{2} \right). $$ Prove that ...
2
votes
0answers
33 views

How to solve the equation $ (x-2)^{\log_{100}(x-2)}+\log_{10}(x-2)^5-12 = 10^{2\log_{10}(x-2)}$?

If $\displaystyle (x-2)^{\log_{10^2}(x-2)}+\log_{10}(x-2)^5-12 = 10^{2.\log_{10}(x-2)}$, then value of $x$ is ... My try:: Let $\log_{10}(x-2) = y\Leftrightarrow (x-2)=10^y$. Then ...
1
vote
0answers
23 views

Deriving cost function using MLE :Why use log function?

I am learning machine learning from Andrew Ng's open-class notes and coursera.org. I am trying to understand how the cost function for the logistic regression is derived. I will start with the cost ...
0
votes
1answer
70 views

Solving the system with logarithms

I tried solving the system $ \begin{cases} (4x)^{\log_2 (2y)} = 64 \\ (8y)^{\log_2 (2y)} = 256 \end{cases} $ several times but still keep getting wrong solutions.
10
votes
5answers
718 views

Solve the equation $2^x=1-x$

Solve the equation: $$2^x=1-x$$ I know this is extremely easy and I know the solution using graphical approach. Basically, I can see the solution, but I can't work it out algebraically.
2
votes
1answer
67 views

Solve for $n$ in $(4/3) ^n = 6439 / 3072 $

It is a series and sequences question. I worked it up to here but I'm stuck at this point. $$\left(\dfrac{4}{3}\right)^n = \dfrac{6439}{3072}$$ Solve for $n$. Thanks
3
votes
2answers
72 views

Solution for this Logarithmic Equation

Recently I was going through a problem from the book Problems in Mathematics - *V Govorov & P Dybov* . $$(x-2)^{\log^2(x-2)+\log(x-2)^5-12}=10^2\log(x-2)$$ I tried solving by first considering ...
1
vote
5answers
79 views

What is the reason to introduce and study logarithmic functions?

I don't understand why logarithms exist when we have exponential functions. Exponential functions seem to be an easier and less convoluted way to write something. Why invent logarithms to do something ...
1
vote
1answer
248 views

Logarithm in an inequality: is it solvable?

Can anyone help me understand what happens to the following inequality once I apply a logarithm to all three parts? $$ - \varepsilon < 2^{\frac{1}{x}} < \varepsilon \longrightarrow \ln(- ...
2
votes
1answer
85 views

How does the Sin and Cos scale on a slide rule work and what is the formula for it?

As I described in this question, I am trying to make a printable slide rule (similar to the slide rule provided in this SciAm article). I have made most of the parts of it but I can't make the Sin and ...
1
vote
0answers
72 views

Approximation for the convolution of normal and lognormal distributions

$$X \sim \ln\mathcal{N}(\mu_X,\,\sigma_X)$$ $$Y \sim \mathcal{N}(0,\,1)$$ $$Z = X + Y$$ I want to find the probability density functions and cumulative distribution functions of $Z$. As the below is ...
4
votes
3answers
51 views

Mechanic method to draw a logarithmic spiral?

I'm in the need to draw (more like to use a wood router to carve a groove) a logarithmic spiral in a piece of wood. So, I got a router that is attached to a stick, I draw a circle by rotating the ...