All Questions
Tagged with logarithms functions
486
questions
-5
votes
2
answers
85
views
If the domain of $f(x)$ is $(-3, 1)$, then what is the domain of $f(\ln x)$? [closed]
I need a clear explanation for this question:
If the domain of $f(x)$ is $(-3, 1)$ then the domain of $f(\ln x)$ is ...
a) $\;(e^{-1}, e^3)$
b) $\;(0, \infty)$
c) $\;(1, \infty)$
d) $\;(e^{-3}, e^...
1
vote
1
answer
66
views
Construction of discontinuous $f$ such that $f(xy) = f(x)+f(y)$ [duplicate]
Question
How to construct a discontinuous $f$ such that $f(xy) = f(x)+f(y)$. Domain of $f$ has to be some subset of $\mathbb{R}$ and range of $f$ is $\mathbb{R}$. Also, try to construct non ...
- 1
1
vote
2
answers
72
views
Log X to what base n yields a whole number [closed]
Does there always exist a real number 'n' such that $log_{n}x$ is a whole number for any real number x?
If yes what would the function to find this number look like?
- 11
0
votes
1
answer
33
views
Natural Log's Property Doesn't Transfer Over
I am trying to rewrite the summation of $\ln(x)$ equation into a continuous function using logarithmic properties. We already know that $\left(\sum_{n=1}^{x}\ln\left(n\right)\right)$ is just equal to $...
- 1
1
vote
2
answers
72
views
Equivalent functions of 8log(2(x)) [closed]
This is an assignment so I'm NOT looking for the answer but please help me understand where I went wrong.
- 15
1
vote
1
answer
99
views
Are there other solutions to the functional equation $f(x^t) = t f(x)$ besides logarithms?
Are there other solutions to the functional equation $f(x^t) = t f(x)$ besides logarithms? Here $x$ and $t$ are real variables with $x>0$.
I know that given the property of logarithms $\log(x^t) = ...
- 1,484
1
vote
2
answers
30
views
Getting the domain of a real function with iterated logarithms [duplicate]
I would like to find the domain of the function
$$f(x)\:=\: \log_4\,\log_5\,\log_3\big(\,18x - x^2 - 77\,\big)$$
as a subset of $\mathbb R\,$.
I looked at the solution of the above problem, and it ...
- 356
2
votes
4
answers
136
views
Solve the equation $\left(\frac{1+\sqrt{1-x^2}}{2}\right)^{\sqrt{1-x}} = (\sqrt{1-x})^{\sqrt{1-x}+\sqrt{1+x}}$
Solve in $\mathbb{R}$:
$
\left(\frac{1+\sqrt{1-x^2}}{2}\right)^{\sqrt{1-x}} = (\sqrt{1-x})^{\sqrt{1-x}+\sqrt{1+x}}
$
My approach:
Let $a = \sqrt{1-x}$ and $b = \sqrt{1+x}$ so $a^2 + b^2 = 2$. The ...
- 637
3
votes
6
answers
360
views
An attempt for approximating the logarithm function $\ln(x)$: Could be extended for big numbers?
An attempt for approximating the logarithm function $\ln(x)$: Could be extended for big numbers?
PS: Thanks everyone for your comments and interesting answers showing how currently the logarithm ...
- 1,484
0
votes
1
answer
42
views
Does the domain of the real logarithm need to be "extended" when dealing with even exponents in the argument?
Consider $f(x)=\log(x^2)$. Clearly, the domain of $f$ is $D_f=\mathbb{R}-\{0\}$, since $x^2>0$ for any $x<0$. However, by the fundamental properties of logarithms:
$$
f(x)=\log(x^2)=2\log(x)\...
- 67
3
votes
2
answers
78
views
Logarithmic Equation Involving Trigonometric Functions
Solve the following equation in real numbers:
$\log_2(\sin x) + \log_3(\tan x) = \log_4(\cos^2 x) + \log_5(\cot x)$
My approach:
$\log_2(\sin x) + \log_3\left(\frac{\sin x}{\cos x}\right) = \log_2(\...
- 637
0
votes
1
answer
67
views
Solutions to Some Logarithmic Inequalities
Suppose we have an inequation as shown below:$$I_0:\space \ln (x) > \frac{x-2}{x}$$ Now we would like to find the largest set $S$ of real numbers such that any element $p\in S$ will satisfy $I_0$ ...
0
votes
0
answers
33
views
Absolute value in integrals leading to logs
I know that when working with real numbers, the input for a logarithm cannot be negativea, and hence when taking the integral of $\frac{1}{x}$, we take the absolute value:
$$ \int{ \frac{1}{x}dx}=ln|x|...
- 1,834
3
votes
0
answers
78
views
What functions satisfy $f(ax) - f(a(x-1)) > f(b(x+1)) - f(bx)$ for all $a, b \in \mathbb{R}^+$ and $x \in \mathbb{Z}^+$.?
I am looking at a family of functions $f : [0, \infty) \rightarrow [-\infty, \infty)$ satisfying the following property:
$$f(bx) - f(b(x-1)) > f(a(x+1)) - f(ax) \quad \text{for all $a, b \in \...
- 31
0
votes
0
answers
80
views
Proving or disproving a mathematical argument
I have to either prove or disprove a mathematical argument:
$$ \log(f(n)) = o(\log(g(n))\rightarrow f(n) = o(g(n))$$
$small-o$ is an asymptotic notation that is explained below. (the $o$ that appears ...
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