All Questions
161
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
0
votes
0
answers
343
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}$$ Now we would like to find the largest set $S$ of real numbers such that any element $p\in S$ will satisfy $I_0$ ...
0
votes
0
answers
30
views
Show that for $a \neq b$ it holds: $\frac{e^b-e^a}{b-a} < \frac{e^b+e^a}{2}$ [duplicate]
Show that for $a \neq b$ it holds:
$$\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$$
...
0
votes
1
answer
124
views
Solving $\ln\left(\frac{1}{x-2}\right)=\frac{1+2e^x}{e^x}$ [closed]
Here's the question I came across, they're inverses in this case, but I imagine that there is a way to do that without them being inverses.
$$\ln\left(\frac{1}{x-2}\right)=\frac{1+2e^x}{e^x}$$
1
vote
0
answers
88
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Does $\log\left(\frac{A}{B}\right)$ really equal to $\log(A)-\log(B)$? [duplicate]
I was investigating the laws of logarithm and playing with Desmos when I realized something curious.
The example equation is $f(x)=\log\left(\frac{2x-4}{6x-8}\right)$ and the graph is this:
The law ...
- 11
1
vote
3
answers
263
views
Can the "simple" equation $e^x=\log(x)$ be solved using algebra?
I came across this really simple-looking yet astonishingly hard problem to solve:
$$e^x=\log(x).$$
I tried to use Lambert-W function, but I cannot get it to the required standard form. Even Wolfram ...
- 13
0
votes
0
answers
98
views
How to solve the equation ${\log_{x} 9} + {\log_{2} x} = {\log_{2} (9+x)}$
Please advise on how to solve the following equation.
$$
{\log_{x} 9} + {\log_{2} x} = {\log_{2} (9+x)}
$$
I tried to rearrange and got this.
$$
{\log_{x} 9} = {\log_{2} (1+\frac{9}{x})}
$$
$$
[{\log_{...
11
votes
1
answer
3k
views
Why can you not divide both sides of the equation, when working with exponential functions?
We recently started with exponential functions, and I did this task for fun, but I apparently did everything wrong. I just don't get why it is wrong. I am aware of some logarithmic properties like $\...
- 156
2
votes
4
answers
254
views
How to solve $x + 3^{x} = 4$ using Lambert W Function.
As stated in the title I am trying to solve the equation $$x + 3^{x} = 4$$ using Lambert W Function and which led me to the result $$x = 4 - \frac{W(3^{4} \ln{3})}{\ln{3}}$$ and driven by the belief ...
- 193
3
votes
3
answers
194
views
How to find the number of solutions of $(0.01)^x=\log_{0.01}x$?
How to find the number of solutions of $(0.01)^x=\log_{0.01}x$?
I drew the graph of $a^x$ and $\log_ax$, with $0<a<1$, and thought they intersect just once.
But the answer given is $3$.
Wolfram ...
- 8,338
0
votes
1
answer
48
views
How to simplify $\sum_{k=0}^{n} \frac{e^{\sum_{i=0}^kx_{i}}}{\sum_{i=0}^k x_{i}}$
Is it possible to simplify
$$
S(\mathbf{x})=\sum_{k=0}^{n} \frac{e^{\sum_{i=0}^kx_{i}}}{\sum_{i=0}^k x_{i}}
$$
A few observations:
$\sum_{k=0}^{n}\sum_{i=0}^k x_{i}=\sum_{k=0}^{n}(n+1-k)x_k$
$ e^{\...
- 3,435
1
vote
1
answer
88
views
Isolating variable in exponents in a linear combination of exponential functions
I'm doing a physics problem that's asking me to compare the maximum speeds of a simple harmonic oscillator versus one immersed in a fluid that's leading to overdamping. I am at a point where I have ...
- 2,220
-1
votes
1
answer
79
views
All real and closed-form roots of $\log_2x=\frac {2^{x-1}}{x}$
What are the closed-form roots of
$$\log_2x=\frac {2^{x-1}}{x}$$
?
My attempts:
Closed-form means, I assume that the Lambert W function can work.
I know that, at least $x>0$.
Wolfram Alpha gives ...
- 1,659
5
votes
3
answers
347
views
Can we express the value of $b^a$ in terms of $c$ , where $c=a^b$?
We know that ;
If $a+b = c$, then $b+a = c$
If $a-b = c$ , then $b-a=-c$
If $ab = c$ then $ba = c$
If $\dfrac{a}{b} = c$ then $\dfrac{b}{a} = \dfrac{1}{c}$
Now, I am curious to know that If $a^{b} = c$...
- 2,821