All Questions
11
questions
-2
votes
1
answer
62
views
What is the product of the solutions to the equation (√10)(x^(log (x))) = x^2? [closed]
I have spent a few hours on this question but I can't seem to grasp it. I have found that taking the log of both sides doesn't give me any progress. The log exponent identity didn't work. I also tried ...
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
5
votes
5
answers
435
views
Contest Math Question: simplifying logarithm expression further
I am working on AoPS Vol. 2 exercises in Chapter 1 and attempting to solve the below problem:
Given that $\log_{4n} 40\sqrt{3} = \log_{3n} 45$, find $n^3$ (MA$\Theta$ 1991).
My approach is to ...
- 682
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
3
votes
3
answers
260
views
Find the integer root of the logarithmic equation: $\log_2\left(\frac x5\right)+\log_x\left(\frac 25\right)=\log 4$
This is a contest question asked 6-7 months ago in a real-time mathematics competition with high-school students. Unfortunately, since the source is not in English, I cannot add it here.
Find the ...
- 111
2
votes
3
answers
196
views
$2+\log _{2} a=3+\log _{3} b=\log _{6}(a+b)$
The problem
Given that $a,b>0$ and $$2+\log _{2} a=3+\log _{3} b=\log _{6}(a+b)$$
Find the value of $$\log _{a b}\left(\frac{1}{a}+\frac{1}{b}\right)$$
My attempt
We have from the given ...
- 10.2k
4
votes
4
answers
633
views
(AIME 1994) $ \lfloor \log_2 1 \rfloor + \lfloor \log_2 2 \rfloor + \ldots + \lfloor \log_2 n \rfloor = 1994 $
$($AIME $1994)$ Find the positive integer $n$ for which
$$ \lfloor \log_2 1 \rfloor + \lfloor \log_2 2 \rfloor + \lfloor \log_2 3 \rfloor + \ldots + \lfloor \log_2 n \rfloor = 1994 $$ where $\lfloor ...
- 41
0
votes
1
answer
405
views
Logarithm in the exponent
$$(2x)^{\log 2} = (3y)^{\log 3} \\
3^{\log x} = 2^{\log y}$$
Solve for $x$ and $y$.
My intuition for solving such problems is taking the logarithm on both sides but it does not work. I also ...
1
vote
2
answers
114
views
Find all values of $x$
Determine all real values of $x$ such that: $$\log_{2}(2^{x-1} + 3^{x+1}) = 2x - \log_{2}(3^x) $$
Let $u = 2^x$ and let $y = 3^x$
For ease, let $\log_{2}$ be represented by just $\log$ so:
Then, $\...
- 11.2k
1
vote
2
answers
150
views
How many pairs $(m, n)$ exist?
For certain pairs $ (m,n)$ of positive integers with $ m\ge n$ there are exactly $ 50$ distinct positive integers $ k$ such that $ |\log m - \log k| < \log n$. Find the sum of all possible values ...
- 11.2k
19
votes
10
answers
1k
views
Find the integer closest to $\ln(2013)$
I encounter such a problem, in a Maths contest, to find out the closest integer to $\ln(2013)$, without using a calculator. I really get stuck.
I tried to turn $\ln(2013)$ into $\ln(3)+\ln(11)+\ln(61)...
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