All Questions
15
questions
-2
votes
1
answer
338
views
How do you solve: $\log_{\cos(x)} \sin(x) + \log_{\sin(x)} \cos(x) \le 2$ [closed]
I have the following inequation:
$$\log_{\cos(x)} \sin(x) + \log_{\sin(x)} \cos(x) \le 2$$
I know that $\sin(x)$ and $\cos(x)$ will give values in the interval $[-1, 1 ]$ but in the base there can't ...
- 1
1
vote
2
answers
112
views
Solving $\sin(x) + \ln(x) = 0$ without a calculator
How would you algebraically handle
$$\sin(x) + \ln(x) = 0$$
to find the zeroes without a graphing calculator?
Here's the graph
Is approximation your only hope? Can you get an exact symbolic answer?
- 913
1
vote
2
answers
146
views
Logarithm to trigonometry conversion
I have an expression like
$$\frac{1}{2} i \log \left(\frac{a-b+i c}{a+b-i c}\right)$$
I was wondering if it is possible to write it in terms of trig functions. I guess it is possible to write all logs ...
- 712
0
votes
1
answer
87
views
Evaluating $2\log_{4} (\cos 67^\circ) + \log_{2}(\cos 23^\circ) + \log_{1/2}(\cos 44^\circ)$
How to calculate the following?
$$2\log_{4} (\cos 67^\circ) + \log_{2}(\cos 23^\circ) + \log_{1/2}(\cos 44^\circ)$$
- 11
1
vote
3
answers
60
views
Proving that $f(x) = 6\ln(x^{11}-4) -2$ is one-to-one
Please verify my proof, and if there are any mistakes please explain.
Prove that that this function is one-to-one: $f(x) = 6\ln(x^{11}-4) -2$.
Suppose $f(x_1) = f(x_2)$
$\implies 6\ln(x_1^{11}-4) -2 =...
1
vote
1
answer
169
views
Domain of $f(x) = \log(\tan x - \sqrt{3})$
This is my attempted solution for finding the domain of $f(x) = \log(\tan x - \sqrt{3})$:
$$
D(f)=\left(\frac{-\pi}{2},\frac{-\pi}{3}\right]\cup\left[\frac{\pi}{3},\frac{\pi}{2}\right)$$
I got this ...
- 99
3
votes
0
answers
136
views
Find value of $\int_{0}^{\frac{\pi}{2}} \log ^2(\sin x)dx$ [duplicate]
Find value of $$I=\int_{0}^{\frac{\pi}{2}} \log ^2(\sin x)dx \tag{1}$$
I tried using property $\int_{a}^{b} f(x)dx=\int_{a}^{b}f(a+b-x)dx$
we get
$$I=\int_{0}^{\frac{\pi}{2}}\log^2(\cos x)dx \tag{2}$...
- 10.2k
2
votes
2
answers
102
views
solve equation with cos and powers
Solve the following equation.
$$3^{x^2-6x+11}=8+\cos^2\frac{\pi x}{3}.$$
Here is the problem that I am struggling with it
I tried to take a logarithm of both side but I kinda stuck
can someone help ...
1
vote
2
answers
48
views
Natural logarithmic function properties
I'm just curious if $-\ln(\cos(x))$ is equivalent to $\ln((\cos(x))^{-1})$? I know the properties of the natural logarithmic function. I just don't know if the property also implies the signs.
...
- 63
1
vote
2
answers
377
views
I need to create a logarithmic equation with 3 points known. [closed]
I need to create a logarithmic equation that intersects with the points:
$(1,504), (20,803.25), (500,7526.925)$
It should be something like this: $$y=\ln(x)+504$$
There must be some extra variables ...
- 43
1
vote
0
answers
56
views
Solve the Logarithmic equation in $[0 \:\: 2\pi]$
Find number of solutions of the Logarithmic equation in $[0 \:\: 2\pi]$
$$\log_{\sqrt{5}}\left(\tan x\right)+ \log_{\sqrt{5}}\sqrt{\log_{\tan x}(5\sqrt{5})+\log_{\sqrt{5}}(5\sqrt{5})}=-\sqrt{6}$$
i ...
- 10.2k
1
vote
3
answers
417
views
Find all possible values of $x$ if $\ln(x) = \sin(x)$.
My Mathematics Teacher gave me the following problem :
Find all the possible values of $x$ if $\ln(x) = \sin(x)$.
I tried graphing both $\ln(x)$ and $\sin(x)$. I found that they intersect at a ...
user399078
0
votes
1
answer
18
views
How do I simplify this expression to a single logarithm involving two -log?
I have to find x in this equation. This is my first time encountering an expression with two -logs so I don't know which one goes in the denominator. The expression is:
$$\log_7(x+6) - \log_7(x-9) - \...
- 55
1
vote
2
answers
86
views
Find all solutions of $\left[\ln(\sin^{-1}(e^x))\right]^5=\ln(\sin^{-1}(e^x))$
The question is: Find all solutions of $\left[\ln(\sin^{-1}(e^x))\right]^5=\ln(\sin^{-1}(e^x))$, where $x$ is real.
Give the solutions in exact form.
What I have done
$$\left[\ln(\sin^{-1}...
- 5,637
7
votes
3
answers
2k
views
Is there a trick to establish $z = \tan \left[ \frac{1}{i} \log \left( \sqrt{ \frac{1+iz}{1-iz} } \right) \right]$?
Im trying to show that
$$ z = \tan \left[ \frac{1}{i} \log \left( \sqrt{ \frac{1+iz}{1-iz} } \right) \right] $$
My first thought is to use the fact that $\sin x = \frac{ e^{ix} - e^{-ix} }{2i } $ ...
user139708