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Algebra II

How to find the limit of $\frac{\ln (\ln (\frac{n}{n - 1}))}{\ln (n)}$ ?
How do you find
$lim_{n \to \infty} \frac{\ln (\ln (\frac{n}{n - 1}))}{\ln (n)}$
I know it's -1, but I had to plot it.

eszkortwxp 2022-04-20

Solving
$\log_{2} (x - 4) + \log_{2} (x + 2) = 4$
Here is how I have worked it out so far:
$\log_{2} (x - 4) + \log (x + 2) = 4$
$\log_{2} ((x - 4) (x + 2)) = 4$
$(x - 4) (x + 2) = 2^{4}$
$(x - 4) (x + 2) = 16$
How do I proceed from here?
$x^{2} + 2 x - 8 = 16$
$x^{2} + 2 x = 24$
$x (x + 2) = 24$ Which I know is not the right answer
$x^{2} + 2 x - 24 = 0$ Can't factor this

Halle Marsh 2022-04-20

Solve ${(\log_{3} x)}^{2} + \log_{3} (x^{2}) + 1 = 0$

Hugh Walls 2022-04-20

Solve
$\log x + \log (2 x - 5) = \log 96 - \log 8$
I started by doing this:
$\log 2 x^{2} - 5 x = \log 12$
Then I got rid of the logs. Can I do that?
$2 x^{2} - 5 x = 12$
What do I do next? What is x?

Molecca89g 2022-04-19

Solve for x: question on logarithms.
The question:
$\log_{3} x \cdot \log_{4} x \cdot \log_{5} x = \log_{3} x \cdot \log_{4} x \cdot \log_{5} x \cdot \log_{5} x \cdot \log_{4} x \cdot \log_{3} x$
My mother who's a math teacher was asked this by one of her students, and she can't quite figure it out. Anyone got any ideas?

painter555ui8n 2022-04-18

How do I solve for V in this equation?
$\frac{\ln (T + 1) - \ln (V - S + 1)}{\ln (V - S + 1)} = \frac{1}{K} - 1$
What are the steps if I want to solve for V?

Malachi Mullins 2022-04-17

How do I prove the following: if $a + \sqrt{m} = b + \sqrt{n}; then a = b and m = n (\sqrt{m} \in R; \sqrt{n} \in R)$ ?

Bryanna Leon 2022-04-16

Is $\log_{23} > \log_{35}$ ?

Camila Glenn 2022-04-16

What is $\frac{d y}{d x}$ of a square root using logarithmic differentiation?
I thought I knew how to do logarithmic differentiation until I came across this:
$y = \sqrt{x (x + 7)}$ find dy/dx by logarithmic differentiation
I was not phased by the problem and did the following work:
$\ln (y) = \ln (\sqrt{x (x + 7)})$
$\ln (y) = \frac{1}{2} (\ln (x) + \ln (x + 7))$
$\frac{1}{y} \frac{d y}{d x} = \frac{1}{2} (\frac{1}{x} + \frac{1}{x + 7})$
$\frac{d y}{d x} = \frac{y}{2} (\frac{2 x + 7}{x^{2} + 7 x})$
but then found out that that was not actually what the answer was if you take dy/dx without logarithmic differentiation. What am I doing wrong? I have spent the past hour trying to figure it out with no luck.

Braden Hatfield 2022-04-16

How do I solve this equation involving a logarithm?
$x \log (x) = 100$

Rubi Riggs 2022-04-16

How do I put $\sin (- 3 \frac{π}{4})$ in radical form?

Kennedy Ford 2022-04-15

Is $1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \frac{1}{5} - \dots = \ln 2$ true?

Breanna Fisher 2022-04-15

Is $lim_{x \to \infty} \frac{\ln {x}}{x} = lim_{x \to \infty} \frac{\frac{1}{x}}{1}$
In my notes its given $lim_{x \to \infty} \frac{\ln {x}}{x} = lim_{x \to \infty} \frac{\frac{1}{x}}{1}$
Is that correct? How do I get that?
I think another example is also related
$x^{\frac{1}{x}} = e^{\frac{\ln {x}}{x}}$

Saul Cochran 2022-04-15

For what values of m and n does $\int_{C} z^{m} z^{- n} d z = 0$ and for what values does $\int_{C} z^{m} z^{- n} d z = 2 i π$ ?
I am stuck on this problem, any hint?

Leonard Montes 2022-04-15

Find roots of a complex variable such that $1 + ω^{m} + ω^{2 m} + \dots + ω^{(n - 1) m} = 0$
I need some help understanding the intuition behind the following question:
Consider the root of $z^{n} = 1$ given by $ω = \cos \frac{2 π}{n} + i \sin \frac{2 π}{n}$ . For which integers m is $1 + ω^{m} + ω^{2 m} + \dots + ω^{(n - 1) m} = 0$ ?

Emmy Decker 2022-04-15

How do you determine all of the pairs of natural numbers x,y radical so that $\sqrt{x} + \frac{2005}{\sqrt{y}} = 2006$ ?

Devin Dougherty 2022-04-14

Lambert W / Product log function?
I would like to solve this equation:
$n \cdot 2^{n} = 15000$
And according to WolframAlpha
$n = \frac{W (15000 \log (2))}{\log (2)}, where \log is ln$
Which shows that I need to use the product log function W which I tried looking up on wikipedia. I don't need the complex numbers, just real numbers.
Additionally, are there ways of solving the original equation without the W functions?
Can someone explain the rules and how to do this please?
I would eventually like to implement a way to find n programmaticly (if possible-in python).

Brielle James 2022-04-14

Differentiate $y = \sqrt{\frac{1 + 2 x}{1 - 2 x}}$ logarithmically
$y = \sqrt{\frac{1 + 2 x}{1 - 2 x}}$
$\ln y = \frac{1}{2} \ln (1 + 2 x) - \frac{1}{2} \ln (1 - 2 x)$
$1 / y d y / d x = 1 / 2 \times 2 / (1 + 2 x) - 1 / 2 \times - 2 / (1 - 2 x)$
$1 / y d y / d x = 1 / (1 + 2 x) + 1 / (1 - 2 x)$
$d y / d x = (1 - 2 x) + (1 + 2 x) / (1 + 2 x) (1 - 2 x) \times y$
$d y / d x = 2 {(1 + 2 x)}^{1 / 2} / (1 + 2 x) (1 - 2 x) {(1 - 2 x)}^{1 / 2}$
$d y / d x = 2 / {(1 + 2 x)}^{1 / 2} {(1 - 2 x)}^{3 / 2}$
However the answer is:
$d y / d x = 2 / (1 - 2 x) \sqrt{(1 - 4 x^{2})}$
Where have I went wrong?

Axel Stout 2022-04-13

Manipulating log function
I'm trying to find the value of a constant for $y (0) = 0$ in the following differential equation.
$2 \ln (2 x + 3 y - 1) - \frac{2 x + 3 y}{2} = 2 x + 3 y + k$
Of course when plugging in the values, I get $2 \ln (- 1) = k$ which errors. When entering this into Wolfram Alpha, they suggest rearranging the equation from this format,
$2 \ln (x - 1) - \frac{x}{2} + constant$
to
$- \frac{x}{2} + 2 \ln (1 - x) + \frac{1}{2} + constant$
"Which is the equivalent for restricted x values", which indeed I have. This would leave me with a positive $\ln (1)$ , which would solve my problem... but...
My question, how does this manipulation work? I've never seen this before. I don't understand how they've made that leap (or if it's even accurate).
Can anyone educate me?

Emmy Decker 2022-04-13

Is this log form simple enough?
$\frac{3}{\ln 2 - 12}$
Is this form simplified enough?
There is a number '12' below the fraction line, do i need to transform the log more to make it simpler?
I wrote that in a college math exam

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