All Questions
117
questions
0
votes
6
answers
195
views
How would you prove $\log_{2}x < \sqrt x$ for $x > 16$? [closed]
I'm not really showing how to prove this, since I tried finding the $x$-intercepts/zeros of $f(x) = \sqrt x - \log_{2} x$ , and see that $x = 4, 16$ work but inspection, but I'm not sure how to ensure ...
- 883
1
vote
3
answers
115
views
How to solve $x+1=5e^{4x}$ [closed]
How to solve $x+1=5e^{4x}$
In general, I know to take ln() of both sides to bring down the exponent for e, but the left side is also a variable.
- 4,771
-2
votes
1
answer
59
views
How does $\log(y)=C+t$ become $y = C e^{t}$? [closed]
I came across this transformation :
$$\begin{align}
\log(y) &= C + t \tag{1} \\[4pt]
y &= C e^{t} \tag{2}
\end{align}$$
How was the first step simplified into the second?
- 7
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 $$x^9=3^x\\(x^9)^{\dfrac {1}{9x}}=(3^x)^{\dfrac {1}{9x}}\\x^ { \ \frac 1x}=3^{ \ \frac 19}\\x^ { \ \...
- 25.2k
1
vote
2
answers
141
views
Simplifying logarithmic expression
Background
An integral is solved, and I get the following expression:
$$I = \frac12 \ln \left| \frac{u-1}{u+1} \right| - \frac u{u^2 - 1} + C$$
Teacher's solution states that from there:
If we set $u ...
- 4,124
3
votes
1
answer
80
views
Calculating (Approximate) values of fractional powered real numbers without calculator or log/antilog tables.
Is there any way to calculate/approximate values like $$(\frac{125}{250})^{0.66}$$ using only a pen, paper and the mind?
(Above expression being just an example, the numbers may vary and not be easy ...
- 35
0
votes
0
answers
51
views
Question about $\log_1 (1)$ [duplicate]
In my college calculus class we just covered properties of logs, and I wanted to ask about them. Two of them are these:
For all $0 < a$: $$\log_a1=0$$
and for all $-\infty\leq b \leq \infty$: $$\...
0
votes
0
answers
63
views
What are some good English mathematics exercises textbooks at the level of South East Asia high school programs?
I live in this region and is studying to retake the national test and I need a big amount of tough exercises to practice but I can't find good documents/ textbooks over here. Any appropriate textbooks ...
user1198194
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
-2
votes
2
answers
80
views
Find the domain of the function $\frac{\sqrt{5-x} - \sqrt{6+x}}{\ln(x^2-1)}$ [closed]
I have troubles finding the domain of this function.
$$f(x) = \dfrac{\sqrt{5-x} - \sqrt{6+x}}{\ln(x^2-1)}$$
Thank you for any help.
Here is what I've done so far:
for $\sqrt{5-x}$, $5\ge x$
for $\...
- 33
0
votes
1
answer
83
views
Why is (log y) = m(log x) + c a straight line? (TMUA question)
In the TMUA (Test of Mathematics for University Admission) specimen paper 1 there is the following question:
The solution provided is as follows:
I am having trouble understanding the opening ...
- 29
0
votes
3
answers
67
views
Question on the natural logarithm laws
Can $3*\ln(x)$ can be written as $\ln(x^3)$ ?
There is a law that $\ln(x^3) = 3\ln(x)$, but does it apply in the reverse case? If not, why not?
- 387
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
1
vote
0
answers
101
views
$\log(x)+\log(y)$ with respect to $x+y$
This is probably a very dumb question, but say that we have the quantity $(x+y)$ which we can access directly, not knowing what $x$ and $y$ is separately. Can we obtain an expression for $\log(x)+\log(...
- 560
0
votes
0
answers
61
views
How to fit a Polynomial to the natural log of itself
Lets say I have a polynomial $a_{N}x^{N}\ ...\ a_{3}x^{3}+a_{2}x^{2}+ax\ +\ c$
Is there any method to compute $ ln(a_{N}x^{N}\ ...\ a_{3}x^{3}+a_{2}x^{2}+ax\ +\ c) $
such that the result is a new ...
- 31