All Questions
Tagged with logarithms exponential-function
906
questions
1
vote
1
answer
41
views
What to consider when taking kth root on both sides of equality
Say I have the following expression:
$10^{l} = a^{k}$
If I take the kth root of both sides, does that mean we get:
$10^{\frac{l}{k}} = a$
We don't have to consider anything with plus or minus?
- 883
11
votes
2
answers
680
views
Implicit function equation $f(x) + \log(f(x)) = x$
Is there a function $f \colon \mathbb{R}_{>0} \to \mathbb{R}_{>0}$ such that
$$
f(x) + \log(f(x)) = x
$$
for all $x \in \mathbb{R}_{>0}$?
I have tried rewriting it as a differential equation ...
- 2,005
2
votes
0
answers
87
views
Does complex log(1) have other values than 0?
I want to determine all values of $$ \left[\log \left(3+2 i^{2}\right)\right]^{1-i} $$
First I simplify to $$\left[\log \left(3-2\right)\right]^{1-i}$$
resulting in $$\left[\log \left(1\right)\right]^{...
- 21
2
votes
4
answers
136
views
Solve the equation $\left(\frac{1+\sqrt{1-x^2}}{2}\right)^{\sqrt{1-x}} = (\sqrt{1-x})^{\sqrt{1-x}+\sqrt{1+x}}$
Solve in $\mathbb{R}$:
$
\left(\frac{1+\sqrt{1-x^2}}{2}\right)^{\sqrt{1-x}} = (\sqrt{1-x})^{\sqrt{1-x}+\sqrt{1+x}}
$
My approach:
Let $a = \sqrt{1-x}$ and $b = \sqrt{1+x}$ so $a^2 + b^2 = 2$. The ...
- 637
-1
votes
1
answer
26
views
express log-log relationship as an exponential relationship [closed]
I have several log(y) ~ a + b * log(x) models that I want to express as exponential relationships. I know this involves an expoential transformation, but how do I solve it?
Example:
Step 1: log(y) ~ 2 ...
- 109
2
votes
1
answer
103
views
Exponential and Logarithmic Expressions
Solve in $\mathbb{R}$ the following equation:
$
(3^x+2)^{\log_5(3)} + 2 = (5^x-2)^{\log_3(5)}
$
My approach:
$
(3^x+2)^{\log_5(3)} + 2 = (5^x-2)^{\log_3(5)} = t
$
After some simplification, we get:
$
...
- 637
0
votes
0
answers
341
views
An analytic solution to solve $x^9=3^x$
I want to find a way to solve $x^9=3^x$ analytically, for two roots. one of them can be found below $$x^9=3^x\\(x^9)^{\dfrac {1}{9x}}=(3^x)^{\dfrac {1}{9x}}\\x^ { \ \frac 1x}=3^{ \ \frac 19}\\x^ { \ \...
- 25.2k
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 $I_0$ ...
1
vote
1
answer
74
views
System of equations where x and y are real numbers
Solve in $\mathbb{R}^2$ the system of equations:
\begin{aligned}
3^x - \frac{1}{y^2} &= 25 \\\\
\log_9(x) - \log_2(y) &= 1
\end{aligned}
We can rewrite the second equation as
$\log_3(x) + \...
- 637
0
votes
1
answer
96
views
Why isn't there an exclusive hypernym for exponential and logarithmic?
Mathematicians, transcendental is a very broad word. Why isn't there a narrow word just for exponential and logarithmic? Transcendental is compared to algebraic, but there is no hyponym of ...
- 103
1
vote
2
answers
105
views
Why is y=In x the only logarithmic function with a gradient 1 at x=1? [closed]
I know nothing about logarithmic functions, only they are supposed to be inverse to exponentials?
I just need a written out explanation to why only natural logarithmic functions have a gradient of 1 ...
- 19
1
vote
0
answers
167
views
How do you solve this formula $(1 + 0.02X) ^{1/X} = 1.0161$
$X$ is supposed to be the duration of the loan in years.
The number $0.02$ stands for the nominal interest and the $1.0161$ stands for the compound interest + $1$.
The equation can also be written : $(...
- 131
0
votes
0
answers
30
views
Show that for $a \neq b$ it holds: $\frac{e^b-e^a}{b-a} < \frac{e^b+e^a}{2}$ [duplicate]
Show that for $a \neq b$ it holds:
$$\frac{e^b-e^a}{b-a} < \frac{e^b+e^a}{2}$$
My first idea was to rearrange
$$2 \cdot (e^b-e^a) < (b-a)(e^b+e^a)$$
$$2e^b-2e^a < be^b + be^a - ae^b -e^a$$
...
0
votes
1
answer
66
views
Quick doubt on this function domain
$$f(x) = \frac{e^{1-\ln(x-x^2)}}{\ln(1 - e^{x-x^2})}$$
I was solving this, and I found the domain is the emptyset.
Yet, then I checked with Mathematica and it returned me a different domain. When I ...
- 3,482
0
votes
0
answers
24
views
How do I find the logarithm and exponential of a string of powers of x (including non-integer powers) in terms of x?
I am trying to find a way to separate the negative and positive powers of x in the solution to the exponential and logarithm of real powers of x, where $M_L,M_E\subset \Bbb{R}$:
$$y=\ln\left(\sum_{k \...
- 53