All Questions
Tagged with algebra-precalculus logarithms
1,541
questions
74
votes
5
answers
5k
views
A new imaginary number? $x^c = -x$
Being young, I don't have much experience with imaginary numbers outside of the basic usages of $i$. As I was sitting in my high school math class doing logs, I had an idea of something that would ...
- 853
43
votes
7
answers
6k
views
Why is $\log(\sqrt{x^2+1}+x)$ odd?
$$f(x) = \log(\sqrt{x^2+1}+x)$$
I can't figure out, why this function is odd. I mean, of course, its graph shows, it's odd, but when I investigated $f(-x)$, I couldn't find way to $-\log(\sqrt{x^2+1}+...
- 978
40
votes
3
answers
3k
views
How should I be avoiding this mistake? (To avoid missing solutions)
First of all, I am sorry if this is a question too simple or stupid.
Consider the equation:
$$
\log((x+2)^2) = 2 \log(5)
$$
If I apply the logarithm law $ \log_a(b^c) = c \log_a(b) $
$$
\begin{...
- 1,107
35
votes
10
answers
4k
views
If both $a,b>0$, then $a^ab^b \ge a^bb^a$ [closed]
Prove that $a^a \ b^b \ge a^b \ b^a$, if both $a$ and $b$ are positive.
- 439
31
votes
9
answers
11k
views
Underlying Reason For Taking Log Base 10
For the equation $2^x = 7$
The textbook says to use log base ten to solve it like this $\log 2^x = \log 7$.
I then re-arrange it so that it reads $x \log 2 = \log 7$ then divide the RHS by $\log 2$ ...
- 441
19
votes
10
answers
1k
views
Find the integer closest to $\ln(2013)$
I encounter such a problem, in a Maths contest, to find out the closest integer to $\ln(2013)$, without using a calculator. I really get stuck.
I tried to turn $\ln(2013)$ into $\ln(3)+\ln(11)+\ln(61)...
- 2,201
18
votes
4
answers
165k
views
How to figure out the log of a number without a calculator?
I have seen people look at log (several digit number) and rattle off the first couple of digits.
I can get the value for small values (aka the popular or easy to know roots), but is there a formula. ...
- 7,362
17
votes
1
answer
744
views
Interesting negative decimal number notation
I was studying logarithms, and had to solve the problem:
If $\log 8 = 0.90$, find $\log 0.125$.
I found out the answer to be $-0.90$. That was easy. But my text book has given the answer as:
$$-...
- 6,479
16
votes
7
answers
4k
views
Smallest Possible Power
When working on improving my skills with indices, I came across the following question:
Find the smallest positive integers $m$ and $n$ for which: $12<2^{m/n}<13$
On my first attempt, I ...
- 213
16
votes
9
answers
4k
views
Intuition behind logarithm change of base
I try to understand the actual intuition behind the logarithm properties and came across a post on this site that explains the multiplication and thereby also the division properties very nicely:
...
- 2,493
16
votes
6
answers
663
views
$\log_9 71$ or $\log_8 61$
I am trying to know which one is bigger :$$\log_9 71$$ or $$\log_8 61$$ how can i know without using a calculator ?
- 161
16
votes
3
answers
619
views
How to prove $\sqrt[\pi]{e} < \sqrt[\pi]{\pi}<\sqrt[e]{e}< \sqrt[e]{\pi}$
I was given a challenge of sorting the following numbers. $\Large\sqrt[\pi]{e} < \sqrt[\pi]{\pi}<\sqrt[e]{e}< \sqrt[e]{\pi}$. After some work I was able to figure out the order. How can one ...
- 11k
16
votes
1
answer
1k
views
Are Base Ten Logarithms Relics?
Just interested in your thoughts regarding the contention that
the pre-eminence of base ten logarithms is a relic from
pre-calculator days.
Firstly I understand that finding the (base-10) ...
- 7,789
15
votes
7
answers
2k
views
Given $\frac{\log x}{b-c}=\frac{\log y}{c-a}=\frac{\log z}{a-b}$ show that $x^{b+c-a}\cdot y^{c+a-b}\cdot z^{a+b-c} = 1$
Given:
$$\dfrac{\log x}{b-c}=\dfrac{\log y}{c-a}=\dfrac{\log z}{a-b}$$
We have to show that :
$$x^{b+c-a}\cdot y^{c+a-b}\cdot z^{a+b-c} = 1$$
I made three equations using cross multiplication :
$$...
- 3,517
15
votes
4
answers
293
views
High School Advanced Functions: Clarifying log rules in a log equation - $\log(x^2) = 2$, Solve for x.
I got in an argument with my teacher for the possible solutions of x. From some sources i found that because x is squared, negative values should be possible; however, my teacher insists that:
$$
\log(...
- 151