All Questions
Tagged with logarithms exponentiation
441
questions
-2
votes
0
answers
141
views
Solving $\sqrt{x+1}=-2$ and looking for complex solutions [duplicate]
So the question was
$$\sqrt{x+1}=-2$$
And obviously there is no value for it,
However,
If you do the thing with $e$ and $\ln{}$
$$e^{\ln{\sqrt{x+1}}}$$
and
$$e^{\frac{1}{2}\cdot (\ln{x+1})}$$
Then ...
- 19
2
votes
2
answers
62
views
Why the property of exponents holds true even for fractional powers
How can we prove that the property of exponents a^m × a^n =a^(m+n) (where "^" this sign denotes the power a is raised to)holds true even if m, n and a are fractions? Like I can clearly see ...
- 49
0
votes
0
answers
34
views
Why is there no logarithmic form of the exponential distributive rule/power of a product rule?
When learning the laws of exponents and logarithms, one finds that there is a correspondence. Each law of exponents has a corresponding equivalent expression in terms of logarithms. For example, the ...
- 16.9k
1
vote
4
answers
922
views
Why roots aren't the inverse of exponentiation but logarithms?
I think it's easy to see it when we look at the inverse of the function "$f(x) = a^x$" but I wonder if there's other way to look at it besides just analyzing the function. I was taught my ...
- 21
2
votes
0
answers
74
views
Principal branch of $z^{1-i}$
I am solving a problem
Find the principal branch of $z^{1-i}$.
I wanted to verify my solution.
I know we can write $z^{1-i} = e^{(1-i)\cdot \text{Log}_e(z)}$
Since the principal branch of $\text{Log}...
- 148
3
votes
2
answers
77
views
How to evaluate an expression of higher powers and roots using logarithms?
I am struggling with the following question from a Dutch algebra exam from the 1950s. The instructions are as follows:
Calculate with logarithms.
$$
x = \frac{\sqrt[3]{(23.57^2 - 15.63^2)}}{{0....
-2
votes
1
answer
59
views
How does $\log(y)=C+t$ become $y = C e^{t}$? [closed]
I came across this transformation :
$$\begin{align}
\log(y) &= C + t \tag{1} \\[4pt]
y &= C e^{t} \tag{2}
\end{align}$$
How was the first step simplified into the second?
- 7
0
votes
2
answers
69
views
Suppose a colony of cells starts with 10 cells, and their number triples every hour. After how many hours will there be 500 cells?
I thought it would be log(500), which gives approximately 2.69897. I know that there could be alternative forms of the answer, but for the life of me, I don't understand how they arrive at this ...
- 83
1
vote
3
answers
131
views
Solve $x^2-2x+1=\log_2( \frac{x+1}{x^2+1})$
Solve in $\mathbb R$ the following equation $$x^2-2x+1=\log_2 (\frac{x+1}{x^2+1})$$
First of all from the existence conditions of the logarithm, we have $x > -1$. Analyzing $x^2 - 2x - 1$ , we get ...
- 125
0
votes
1
answer
78
views
$\log_{2} \frac{1-ab}{a + b} = 2ab + a + b -3$. Find min of $(a +b)$
Problem: $a$, $b$ are positive real numbers and $\log_{2} \frac{1-ab}{a + b} = 2ab + a + b -3$. Find the minimum value of $P = a + b$
I reached this point and had no idea how to proceed. Please ...
- 1,426
1
vote
2
answers
97
views
Given $x^2y = 32$ and $x^3/y = 1/8$, find $\log_2 x$ and $\log_4 y$ [closed]
Let $\log_2 x=a$ and $\log_4 y = b$. Then from
$$x^2y = 32$$
and
$$\frac{x^3}{y} = \frac{1}{8}$$
we need to find the values of $a$ and $b$.
I substituted values for $x$ and $y$:
$\log_2 2^2 = 2$ and $\...
0
votes
1
answer
72
views
How is $\log _{10}(e)=\left[\log _e(10)\right]^{-1}$? [duplicate]
I am watching a logarithm lecture from 3Blue1Brown (great for math dummies like me!)
Here is the video for context:
https://youtu.be/4PDoT7jtxmw?t=1306
The step that I did not follow is when he ...
- 491
1
vote
0
answers
111
views
Help Solving a logarithmic equation $P\times\log{(1-\frac{a}{nP})} = -b\times\log{(1+\frac{c}{n})}$ for P where P>0
I have tried using algebraic Logarithm and exponent rules but I cannot get P into a common form. I get P in exponent and standard form or I get P in Logarithmic and standard form
My attempt so far:
$...
- 11
1
vote
1
answer
125
views
What is this "periodic" sequence called?
I came across this weird "periodic" sequence (containing only natural numbers) where the first 15 elements are
$$1, 1, 2, 1, 2, 3, 4, 1, 2, 3, 4, 5, 6, 7, 8.$$
The sequence is not finite, ...
- 31
0
votes
1
answer
102
views
Is it possible to prove that $(x+1)\cdot\ln(x-1) > x\cdot\ln(x)$ for all integer values of $x>4$?
Is it possible to prove that $(x+1)\cdot\ln(x-1) > x\cdot\ln(x)$ for all integer values of $x>4$?
It comes from a broader question about $(x-1)^{x+1} > x^x$ and the logarithm approach seems ...
