All Questions
16
questions
-3
votes
2
answers
191
views
How do you solve this equation $ \log_{2}(x) = \sqrt x$? [closed]
Disclaimer: Guys before voting to get the question closed I strongly feel we should instead have a feature on MSE that can merge such similar/duplicate questions since we got some really cool/through ...
- 883
0
votes
3
answers
58
views
How do you find the exact value of a logarithm with a radical in the base?
I'm struggling to find a method for evaluating $\log_{5\sqrt2} 50$ (or ${\log50}\over{\log5\sqrt2}$) without using a calculator. When using a calculator, I am given an exact value of 2, but I can't ...
4
votes
1
answer
131
views
Showing $ 2\sqrt{\frac{x+3}{x}}+8\sqrt{\frac{x+1}{x}}-\ln\left(\frac{(x+1)^{3/2}(x+3)}{(x-1)^{5/2}}\right)\geq 10 $ for $x\geq7$
Suppose that $x\geq 7$. I would like to show that
$$ 2\sqrt{\frac{x+3}{x}} + 8\sqrt{\frac{x+1}{x}}-\ln\left(\frac{(x+1)^{3/2}(x+3)}{(x-1)^{5/2}}\right)\geq 10 $$
I rewrote the inequality as
$$ 2\sqrt{...
- 1,702
5
votes
1
answer
106
views
Solving $(\sqrt{2})^x+(\sqrt{2})^{x-1}=2(2\sqrt{2}+1)$
I'm in stuck with this simple equation.
$$(\sqrt{2})^x+(\sqrt{2})^{x-1}=2(2\sqrt{2}+1)$$
This is my solution:
$$\begin{align}(\sqrt{2})^x+(\sqrt{2})^x(\sqrt{2})^{-1} &=4\sqrt{2}+2 \tag{1}\\[4pt]
2^...
- 81
0
votes
1
answer
114
views
Solve the equation in real numbers: $x-6+\frac{2}{\sqrt{x-2}}=\frac{1}{3}\log_3(\frac{x}{x^3+54})$
Solve the equation in real numbers: $x-6+\frac{2}{\sqrt{x-2}}=\frac{1}{3}\log_3(\frac{x}{x^3+54})$
My work: I have managed to find that $3$ is a solution to the problem.I tried to prove that this is ...
- 1,144
0
votes
1
answer
90
views
Restrictions on exponential
This question has already been asked but no one answered so:
so i want to add restrictions to this list of exponent laws. The code i will post now is my attempt at writing their restrictions and the ...
- 77
0
votes
1
answer
130
views
Restrictions on exponent laws
Alright, so i want to add restrictions to this list of exponent laws. The code i will post now is my attempt at writing their restrictions and the laws i want to write restrictions on. SO, please let ...
- 77
0
votes
1
answer
36
views
Previously adding restrictions
Okay, being completely honest, i don't know how more to make it clearer, this question has been deleted 3 times, maybe people don't actually read what i say at the beginning, which said perfectly what ...
- 77
3
votes
4
answers
213
views
Comparing $\ln 1000$, $\sqrt[5]{1000}$, $3^{1000}$, and $1000^{15}$ without calculator
In my Pre-Calculus class we were given the following problem:
Put the following four values in order from smallest to largest: $\ln 1000$, principal $5$th root of $1000$, $3^{1000}$, and $1000^{15}$...
- 830
1
vote
2
answers
97
views
Expanding log problem
I found this site with online problems and answers.
https://courses.lumenlearning.com/waymakercollegealgebra/chapter/expand-and-condense-logarithms/
I've tried several problems and my answer is ...
3
votes
3
answers
347
views
Simplify the expression: $a^{\log {\sqrt \frac bc}}×b^{\log {\sqrt \frac ca}}×c^{\log {\sqrt \frac ab}}$
My problem is
Simplify the expression:$$a^{\log {\sqrt \frac bc}}×b^{\log {\sqrt \frac ca}}×c^{\log {\sqrt \frac ab}}$$
Here $a,b,c \in \mathbb {R^+}$
My way:
$$\begin{cases}
\frac bc=e^x ...
- 429
4
votes
10
answers
2k
views
How to find $x$ given $\log_{9}\left(\frac{1}{\sqrt3}\right) =x$ without a calculator?
I was asked to find $x$ when: $$\log_{9}\left(\frac{1}{\sqrt3}\right) =x$$
Step two may resemble:
$${3}^{2x}=\frac{1}{\sqrt3}$$
I was not allowed a calculator and was told that it was possible. I ...
- 130
-2
votes
1
answer
77
views
How to intuitively deduce the relationship among exponent, log and root?
It would really nice to have pictorial representation of how these functions are related and how each unknown can be derived.
(A2A)
- 107
2
votes
2
answers
114
views
Solve $ 1 - \sqrt{1 - 8\cdot(\log_{1/4}{x})^2} < 3\cdot \log_{1/4}x $
My answer: $2^\frac{-1}{\sqrt{2}} < x < 1$
Textbook answer: $2^\frac{-12}{17} < x < 1$
The only difference between my resolution and the Textbook one is that I solved by saying that
$$\...
2
votes
3
answers
130
views
Simplifying $\log_4(\sqrt{2+\sqrt{3}}+\sqrt{2-\sqrt{3}})$
Simplify$$\log_4(\sqrt{2+\sqrt{3}}+\sqrt{2-\sqrt{3}}) .$$
Can we use the following formula to solve it?$$\sqrt{a+\sqrt{b}}= \sqrt{\frac{{a+\sqrt{a^2-b}}}{2}}$$
Therefore first term will become$$\sqrt{...
- 621