Tagged Questions
5
votes
3answers
220 views
Can $\displaystyle\lim_{h\to 0}\frac{b^h - 1}{h}$ be solved without circular reasoning?
In many places I have read that $$\lim_{h\to 0}\frac{b^h - 1}{h}$$ is by definition $\ln(b)$. Does that mean that this is unsolvable without using that fact or a related/derived one?
I can of course ...
0
votes
1answer
44 views
How do I create an equation that decelerates past a certain value?
Apologies for my lack of pure maths, I am a programmer!
I currently have an equation in code that states that if a number goes below a certain value (in my case, 0.7) then the difference is dampened:
...
1
vote
1answer
53 views
Derive the PDF of the log-normal distribution?
If $X \sim N(0,1)$ and $Y = e^X$, find the PDF of $Y$ using the two methods:
(i) Find the CDF of of $Y$ and then differentiate. Use the notation $\Phi(x)$ and $\phi(x)$ for the CDF and PDF of $X$ ...
0
votes
2answers
41 views
Definition of logarithm in complex domain
My first question is:
What is the proper definition of logarithmic function $f(z)=\ln{z}$.
where $z\in \mathbb{C}$.
quoting Wikipedia.
a complex logarithm function is an "inverse" of the ...
1
vote
1answer
43 views
Solving basic exponential equation with logs
I am having trouble with this grade 12 pre-calc question that I am sure will be elementary to most of you. I understand most of it but I do not understand one of the steps.
These are the steps in my ...
1
vote
1answer
25 views
Is it true that $\int t\frac{dF}{d \ln{t}} d \ln{t}=\int \frac{dF}{dt} dt$
It seems to be true that:
$$\int t\frac{dF}{d \ln{t}} d \ln{t}=\int \frac{dF}{dt} dt$$
For eg., this works with $\frac{dF}{dt}=\frac{1}{2} (\cos(\pi \ln{t})+1)$
But then there must be something ...
1
vote
1answer
63 views
exponential population growth models using $e$?
Im trying to understand this write up [1] of cell population growth models and am confused about the use of natural logarithms. If cells double at a constant rate starting from 1 cell, then their cell ...
0
votes
1answer
57 views
Checking whether answers of logarithmic and exponential equalities are correct.
When you check the answers you get from equalities like for example:
$$ ^2\log(x-2) = 3- ^2\log(x)$$
$$ 4^x = 3 \times 2^x + 10$$
so on and so forth, is it sufficient to do the following:
For the ...
2
votes
3answers
240 views
The relation between an exponential function and a logarithmic function
I have been told multiple times that the logarithmic function is the inverse of the exponential function and vice versa. My question is; what are the implications of this? How can we see that they're ...
13
votes
3answers
321 views
Find $x$ from $3^x\cdot x^3 = 1$
I saw a question on internet, tried to solve but I can't:
\begin{equation}
3^x\cdot x^3 = 1
\end{equation}
I get $\ln$ function and made some equalization and I reached that:
\begin{equation}
...
19
votes
18answers
2k views
How to understand why $x^0 = 1$, where $x$ is any real number?
Alright, so the idea of an exponent, $x$, is that you are multiplying its base by itself $x$ number of times. With base $5$ and $x=3$, we have that $5^3$ = $5 \cdot 5 \cdot 5$
I understand that the ...
0
votes
1answer
190 views
Exponential and power functions through two points
I have a problem where I'm asked to determine the constants of exponential and power functions that go through both points (5, 50) and (10, 1600). I have tried to solve them below, but would ...
1
vote
1answer
39 views
Does the logistic function uniquely satisfy these three conditions?
Given
$$r(t)=\frac{f(t)}{1-F(t)} \tag{Eq. 1}$$
where
$$f(t)=\frac{dF}{dt} \tag{Eq. 2}$$
and the conditions:
$$\lim_{t\rightarrow \infty} r(t)=1 \tag{Eq. 3}$$
$$\lim_{t\rightarrow \infty} F(t)=1 ...
0
votes
1answer
80 views
How do I solve this exponential equation?
$$x = 2^{x-3}$$
Does there exist an analytical solution to this equation? If so, how do I find it?
What if it is changed to an equality?
$$x>2^{x-3}$$
3
votes
3answers
154 views
Solve $3\log_{10}(x-15) = \left(\frac{1}{4}\right)^x$
$$3\log_{10}(x-15) = \left(\frac{1}{4}\right)^x$$
I am completely lost on how to proceed. Could someone explain how to find any real solution to the above equation?
-1
votes
1answer
137 views
How to find asymptotic entire functions?
I want to know how to find analytic functions $f(z)$ that are asymptotic and analytic on and near the real line of functions of the type $\ln(C +\exp(P(z^2)))$ where $C$ is a complex constant and $P$ ...
7
votes
5answers
462 views
Solve the equation $2^x=1-x$
Solve the equation:
$$2^x=1-x$$
I know this is extremely easy and I know the solution using graphical approach. Basically, I can see the solution, but I can't work it out algebraically.
1
vote
2answers
143 views
Simplifying the expression of exponential and logarithms
I want to simplify the following expression.
$$Y=\text{Bottom} + \frac{\text{Top}-\text{Bottom}}{1+10^{((\log EC50-X))}}$$
$\log$ is base of $10$. Some may know that it's a dose response curve, and ...
