All Questions
Tagged with logarithms systems-of-equations
111
questions
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
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
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
0
answers
74
views
How to solve this? $a,b\in R, a+\ln{a}=2, \ln{(1-b)}=b+1$, show value of $a+b$.
How to solve this? $a,b\in \Bbb R, a+\ln a=2, \ln (1-b)=b+1$, show value of $a+b$.
Notice that $\ln (1-b)+(1-b)=2$, we have $1-b=a$.
But I don't understand the reason to do this. Why do I have to ...
- 425
2
votes
2
answers
102
views
What are the number of real solutions (x,y) of the system : $3^x +4^y=13 $ and $\log_3(x)-\log_4(y)=1$
I am trying to find number of solutions $(x,y)$ for :
$$\begin{align} 3^x +4^y &=13 \\ \log_3(x)-\log_4(y)&=1\end{align} $$
At first I tried :
$$\log_3(x)=1+\log_4(y)$$
and then :
$x=3^{\log_4(...
- 59
0
votes
0
answers
72
views
How on earth I am supposed to find an expression for C1 from this?
I am currently studing economics and I encountered this problem that is driving me crazy.
I need to solve for C1 from this set of equations:
$\frac{1}{C_1}= \frac{p}{C_2^h}+\frac{q}{C_2^m}+\frac{z}{...
- 11
1
vote
3
answers
106
views
Solve $(1-(-a*\ln(bx))^n)x=\delta$ for $x$
Should be a simple question, but don't know how to solve this: Given $a,b,x,\delta>0$ with $\ln(bx)<0$, I want to solve $$(1-(-a*\ln(bx))^n)x=\delta$$ for $x$, where $n\in\mathbb N$. How do I ...
- 13.9k
2
votes
6
answers
116
views
How many $(x,y)$ solutions does the system $\begin{cases}3^x+4^y=13 \\ \log_3x - \log_4y=1\end{cases}$ have?
How many $(x,y)$ solutions does the system $\begin{cases}3^x+4^y=13 \\ \log_3x - \log_4y=1\end{cases}$ have?
As I tried to solve this problem, I noticed that there is a single pair $(x,y)$ for which $...
- 1,123
2
votes
1
answer
40
views
Solving a system of equations with logarithms in exponents
I am interested in what approach could be applied to solve such a system:
$$ 2^{x-\log_2{y}} - 8^{x+\log_2{y}} = 5y - 2^{x-\log_2{y}} $$
$$ \frac{2^{x-\log_2{y}}}{8^{x+\log_2{y}}} = \frac{5y}{2^{x-\...
- 123
0
votes
1
answer
97
views
Evaluation of all solutions such that $~ y(x)>0 ~$ for $~ x>0 ~$where $~y(x)~$ is a general solution of $2$nd order linear nonhomogeneous DE
The essential problem statement is shown far below this post with bold italic font.
$$ \text{Evalution of solution}~ y(x) ~ \text{of}~ y''+\sqrt{5}y'-y+2=0 $$
My works
$$\begin{align}
y''+\sqrt{5}...
- 1,519
0
votes
0
answers
134
views
Solve the system of equations.
There was a mistake in the previous question but it is now fixed:
$\log_{10}\dfrac{1}{3}(y + 2) \to \log_{\frac{1}{3}}(y + 2)$
Solve the system of equations for $z\ge0$: $\left\{ {\begin{array}{*{20}{...
- 449
4
votes
1
answer
72
views
Proof of positivity of $~ x+\sqrt{x^2+1} ~$ of $~\operatorname{arsinh}(x)=\operatorname{arcsinh}(x)=\sinh^{-1}(x)= \ln \left( x+\sqrt{x^2+1}\right)$
Proof of positivity of $~ x+\sqrt{x^2+1} ~$
I found this formula appears at $~ \operatorname{arsinh}(x)= \ln \left( x+\sqrt{x^2+1}\right)~$
So, of course this argument inside the natural log function ...
- 1,519
0
votes
0
answers
35
views
How did Pareto derive this equation to represent income distribution and why is the use of logarithm?
Vilfredo Pareto argued that in all countries and times, the distribution of income and wealth is highly skewed, with a few holding most of the wealth. He argued that all observed societies follow a ...
- 1
3
votes
2
answers
95
views
Solve the following system of equations, $\log_{9}x = \log_{12}y = \log_{15}(x + y)$, and more generally, $\log_ax = \log_by = \log_c(x \pm y)$.
Solve the following system of equations $$\log_{9}x = \log_{12}y = \log_{15}(x + y)$$
This isn't really just a question about the above system, but more of a question asking for the general method ...
- 4,246
0
votes
1
answer
11
views
Subproblem of transformation to standard normal distribution from binomial distribution using$~x,n\gg0~$
This post is about binomial distribution with standard normal distribution.
$$h(x):=\ln(x)~~\text{where}~~\underbrace{1\ll x\in\mathbb{N}}_{\text{very large natural number}}~~\text{is held}\tag{1}$$
$$...
- 1,519