All Questions
Tagged with algebra-precalculus logarithms
127
questions with no upvoted or accepted answers
4
votes
1
answer
119
views
Solving equations with logarithmic exponent
I need to solve the equation :
$\ln(x+2)+\ln(5)=\lg(2x+8)$
With the change of base formula we can turn this into:
$\ln(x+2)+\ln(5)=\frac{\ln(2x+8)}{\ln(10)}$
We can also simplify the LHS with the ...
- 41
4
votes
1
answer
642
views
Find integer $n$ that satisfies $(\lg n)^{2^{100}} <\sqrt{n}$ with $n > 2$
If $(\lg n)^{2^{100}} < {n^{1/2}}$, where $\lg$ is the binary logarithm, then
$$(\lg n)^{2^{101}} < n$$
$$2^{101}\lg \lg n < \lg n$$
$$101 < \lg \lg n - \lg \lg \lg n$$
I don't know that ...
- 79
3
votes
0
answers
78
views
What functions satisfy $f(ax) - f(a(x-1)) > f(b(x+1)) - f(bx)$ for all $a, b \in \mathbb{R}^+$ and $x \in \mathbb{Z}^+$.?
I am looking at a family of functions $f : [0, \infty) \rightarrow [-\infty, \infty)$ satisfying the following property:
$$f(bx) - f(b(x-1)) > f(a(x+1)) - f(ax) \quad \text{for all $a, b \in \...
- 31
3
votes
0
answers
66
views
Checking solution of equation with exponentials
I have solved the following equation:
$2^x (2+\sqrt{3})^x - 2(1+\sqrt{3})^x = 2$
My approach:
since $2+\sqrt{3}=\frac{1}{2}(1+\sqrt{3})^2$ it follows that $(2+\sqrt{3})^x=\frac{1}{2^x}(1+\sqrt{3})^{2x}...
- 4,098
3
votes
0
answers
83
views
How can I solve a system of 3 equations that use logs?
I'm having a bit of trouble solving this system of equations. What would be a good way of solving for the three variables in a question like this? I've tried many different ways of substituting or ...
- 31
3
votes
0
answers
86
views
Find value of $k$ such that domain of $f(x)$ is $(-\infty \: 2] \cup [6 \:\infty)$
Find value of $k$ such that domain of $$f(x)=\log_{0.5}\left(\log_{4}\left(\log_3[(x-k)^2]\right)\right)$$ is $(-\infty \: 2] \cup [6 \:\infty)$ where $[.]$ is Greatest integer function.
For outside $...
- 13.2k
3
votes
0
answers
175
views
solve $y = \frac{A }{\frac{B}{\ln(y/y_0)} - 1} \frac{1}{x^2}$
I'm trying to express y as a function of x, using the following equation :
$$ y = \frac{A }{\frac{B}{\ln(y/y_0)} - 1} \frac{1}{x^2} $$
Can anyone help me ? Thanks !
[Edit]
- I originally attempted ...
- 31
3
votes
0
answers
147
views
Tweaking Reddit's Ranking Algorithm
This image explains how Reddit's Ranking algorithm works.
As you know, Reddit is a very high traffic site. Therefore, the post rank decreases quite fast. This algorithm puts emphasis on bringing ...
- 131
2
votes
0
answers
68
views
How to solve the equation $x^{\log_3{(x-1)}} = 3^{(3+\log_3{x})}$ [SOLVED]
I made a mistake copying the question ! it should be $x^{\log_3{(x)}-1} = 3^{(3+\log_3{x})}$.
I'm trying to solve the equation:
$x^{\log_3{(x-1)}} = 3^{(3+\log_3{x})}$
What I tried:
I took the ...
- 23
2
votes
0
answers
39
views
Simplifying $\sum_{k=1}^{\log_{2}{n}}\log_{3}{\frac{n}{2^k}}$ in two ways gives different results
I want to calculate the result of
$$\sum\limits_{k=1}^{\log_{2}{n}}\log_{3}{\frac{n}{2^k}}$$ I used two below approaches. Both approaches are based on $\log A + \log B = \log (A \times B)$ and $\sum\...
- 485
2
votes
0
answers
39
views
Does every basic exponent property have a corresponding logarithm property and vice versa?
Here is what I'm referring to:
For $0 < a \neq 1$, $\,0 < b \neq 1$, and $m,n \in \mathbb{R^+}$, $p = a^m$, $q = a^n,$
$$\begin{align*}
a^0 &= 1 \qquad &\qquad \log_a(1) &= 0 \tag{a}\...
- 287
2
votes
0
answers
92
views
Subtract two long functions, possible shortcut using derivative?
If $f(x)$ is as follows:
$$f(x)=\frac{2-\ln(x+1)}{x^4 +2}$$
what is $f(a+h)-f(a)$?
I started this problem but quickly realized that the calculations were going to get difficult and long. But I noticed ...
2
votes
0
answers
51
views
Finding the domain of $[\log_{10}(\frac{5x - x^2}{4})]^{1/2}$
The solution given by the book is:
I think this is wrong. My solution is:
$\frac{5x - x^2}{4}$ $\ge$ $1$
$5x - x^2$ $\ge$ $4$
$5x - x^2 - 4$ $\ge$ $0$
$(-1)(x-1)(x-4)$ $\ge$ $0$
or
$(x-1)(x-4)$ $\le$ ...
- 425
2
votes
0
answers
27
views
Solving an equation with different bases, each to an unknown power, but with a third value
This is a relatively straightforward question, but neither myself nor my colleagues are able to come up with a tidy solution without brute force:
Represent $x$ algebraically, given the equation
$$4^{...
- 178
2
votes
0
answers
54
views
Is there an easy way to find parameter by three information?
$f(x)=a+b.c^x$ and we know
$$f(0)=15\\
f(2)=30\\f(4)=90$$
so put down point in function and have:
$$a+b.1=15\\a+b.c^2=30\\a+b.c^4=90$$so $$(2)-(1) \to bc^2-b=30-15\\(3)-(2) \to bc^4-bc^2=90-30\\b(c^2-...
- 25.2k