All Questions
23
questions
2
votes
1
answer
108
views
How to solve a quadratic raised to another quadratic?
Question:
Solve for all real $x$ $$(x^2 - 7x + 11)^{(x^2 - 11x + 30)} = 1$$
My thoughts:
My first thought was to take logs where the base polynomial would be the power, so $$\log_{x^2 - 7x + 11} \...
- 1,201
4
votes
4
answers
240
views
Contest Math Question on Logarithms
I am trying to solve a question from the AoPS Vol. 2 book.
The question is as follows:
Suppose the $p$ and $q$ are positive numbers for which:
$$\log_9 p = \log_{12}q = \log_{16}(p+q)$$
What is the ...
- 682
1
vote
1
answer
78
views
Making $-\log(2-4x)\log(x-5)+\log(2x-1)=\log(3-4x)$ into a quadratic equation
How we can work with below equation to get quadratic equation?
$$-\log(2-4x)\log(x-5)+\log(2x-1)=\log(3-4x)$$
No need to get the variable x
- 11
2
votes
1
answer
85
views
How to solve this logarithmic equation with sum of exponential functions?
I come across this logarithmic equation recently (solve for $x \in \mathbb{R}$) :
$$ 2x \geq \log_2 \left( \frac{35}{3} \cdot 6^{x-1} - 2 \cdot 9^{x - \frac{1}{2}} \right)$$
With few quick changes, ...
- 121
0
votes
1
answer
62
views
Find Solution set of $(\log_4x)^2+4\sqrt{(\log_4x)^2-\log_2x-4}=\log_2x+16$
My attempt :
$$\dfrac{1}{4}(\log_2x)^2+4\sqrt{\dfrac{1}{4}(\log_4x)^2-\log_2x-4}=\log_2x+16$$
$$(\log_2x)^2+16\sqrt{\dfrac{1}{4}(\log_2x)^2-\log_2x-4}=\log_2x+16$$
Let $a=\log_2x$
$$a^2+8\sqrt{a^2-4a-...
- 569
3
votes
1
answer
100
views
What are the solutions for the following equation?
I have the following equation:
$\log_{2x}4x+\log_{4x}16x=4$
What are the solutions of this equation?
This is what I did:
Firstly, I applied the following conditions:
$2x>0 \Rightarrow x>0$
...
user592938
4
votes
1
answer
139
views
Logarithmic equation $\log_2(x+4)=\log_{4x+16}8$
So the problem goes:
What is the product of all solutions in the equation
$$\log_2(x+4)=\log_{4x+16}8$$
The solution to this should be $31\over4$, but I got $-14$. This is what I did:
\begin{...
- 861
2
votes
2
answers
73
views
Equation involving the logarithm
Find the set of values of $k$ satisfying this equation for only one real root of $x.$
$$ \log(kx) = 2 \log(x+2)$$
I think that for the sake of satisfying the domain restriction:
$ k \cdot x > 0 $
...
0
votes
1
answer
44
views
How I can solve $\log_{\frac{1}{(2+|x|)}}(5+x^2)=\log_{(3+x^2)}(15+\sqrt x)$ [closed]
I don't know how to solve this. I have tried base change but it's not working. Please provide a solution.
- 315
2
votes
1
answer
510
views
How to solve the equation $\log_2(x-9)+\log_{(2x-18)}6=3$.
Solve the following equation. $$\log_2(x-9)+\log_{(2x-18)}6=3.$$
I tried this way,
\begin{align}
\log_2(x-9)+\log_{(2x-18)}6 & \ = \ 3\\
\log_2(x-9)+\log_{2(x-9)}6 & \ = \ 3\\
\log_2(...
- 117
0
votes
3
answers
210
views
How to solve this logarithmic equation: $ \ x^{3\log^3 x-\big(\frac{2}{3}\big)\log x} = 100 \sqrt[3]{10}\ $
How to solve this?
I am new to logarithms.
$$ \ x^{3\log^3 x-\big(\frac{2}{3}\big)\log x} = 100 \sqrt[3]{10}\ $$
All the logs have base $10$.
- 1,471
0
votes
2
answers
85
views
What do I miss? $\ln(x^2 -4) = \ln(1-4x)$, $x \neq 1$ [closed]
Solve
$$\ln(x^2-4) = \ln(1-4x)$$
=>
1)$$x^2-4 = 1-4x$$
2) $$x_{1/2} = -5, 1$$
But since $\ln(-3)$ is not defined, only $x=-5$ is a solution. Shouldn't this already come out while solving for $x$?
- 197
8
votes
2
answers
3k
views
Find the values of $b$ for which the equation $2\log_{\frac{1}{25}}(bx+28)=-\log_5(12-4x-x^2)$ has only one solution
Find the values of 'b' for which the equation
$$2\log_{\frac{1}{25}}(bx+28)=-\log_5(12-4x-x^2)$$ has only one solution.
=$$-2/2\log_{5}(bx+28)=-\log_5(12-4x-x^2)$$
My try:
After removing the ...
user220382
1
vote
2
answers
783
views
How to solve for log with a number outside?
$$\log_6(4x-10)+1 = \log_6(15x+15)$$
This is a sample problem. I know that when the bases of log are the same, all you have to do is set the parenthesis inside equal to each other.
If the $1$ wasn't ...
- 29
0
votes
2
answers
71
views
Solving a logarithmic equation where the logarhitm is exponentiated
I have troubles solving the following logarthitmic equation.
$$ \ 2(\log_x{\sqrt7})^2-\log_x{\sqrt7}-1 =0 $$
The results are supposed to be $ \ x_1 = {\frac{1}{7}}, x_2 = \sqrt7 $
I have tried ...
- 25