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Algebra II

Simplifying Logarithmic Expression
Compute:
$\frac{1 - \log_{a}^{3} b}{(\log_{a} b + \log_{b} a + 1) \log_{a} \frac{a}{b}}$
I tried to expand it :
$\frac{1 - \log_{a}^{3} b}{(\log_{a} b + \log_{b} a + 1) \log_{a} \frac{a}{b}}$
$= \frac{(1 - \log_{a} b) (\log_{a}^{2} b + \log_{a} b + 1)}{(\log_{a} b + \log_{b} a + 1) (1 - \log_{a} b)}$
$= \frac{(\log_{a}^{2} b + \log_{a} b + 1)}{(\log_{a} b + \log_{b} a + 1)}$
But I got nothing.

Deven Livingston 2022-04-25

Simplify log expression with infinite series $\log x - \sum_{{i = 1}}^{\infty} \frac{x^{i}}{i}$

Molecca89g 2022-04-24

Natural log summation representation
In working through a problem, I've encountered the need to express
$\log n = \sum_{k = 1}^{n - 1} \log (1 + \frac{1}{k})$
where $\log k$ is the natural logarithm of k. I'm fairly certain the above equality holds (please let me know if otherwise), but I'm not quite sure how to derive it. Any help would be greatly appreciated.
So far, I've tried expressing $\log n = \log (n!) - \log ((n - 1)!)$ to no avail.
Many thanks!

Ellie Castro 2022-04-24

if $m = \log_{10} n$ , what other relationship exists between m, 10, and n?

hadaasyj 2022-04-24

Perform the indicated operation with complex numbers. (Simplify your answe completely.)
$(4 - 8 i) (9 + 7 i)$

etudiante9c2 2022-04-24

how to find the value of $\log_{37}$

airesex2 2022-04-24

How to derive the following series expansion for $\ln (\frac{x + 1}{x - 1})$
How to prove :
$\ln (\frac{x + 1}{x - 1}) = 2 [\frac{1}{x} + \frac{1}{3 x^{3}} + \frac{1}{5 x^{5}} + \dots]$
where $| x | > 1$
I am not able to get how to proceed on this one ?

Treboldiqtw 2022-04-23

In a and b, solve for x.
a) $\log x = 5$
b) $\log x = 1.25$

Ellie Castro 2022-04-23

Show that if $z_{1}$ and are complex numbers then
$\overset{―}{\frac{z_{1}}{z_{2}}} = \frac{\overset{―}{z_{1}}}{\overset{―}{z_{2}}}$

Lymnmeatlypamgfm 2022-04-23

If $\sqrt{36} = 6$ is true, then why is $\sqrt{36} = - 6$ false in verification of radical equations?

vacinammo288 2022-04-23

If I knew that by the year 2000 that there were 1 trillion humans, and that they started reproducing 2000 years before that, how would I calculate their birth rate?
Assume an initial population of 1 million, and a reproduction age of 15.
Also take into account a standard death rate, where death occurs after 75 years, and reproduction age ends at 50.
Assume a uniform age distribution.

Averie Ray 2022-04-23

How should I calculate the Lebesgue integral of logarithm function from zero to infinity?
Does the area under the $\ln (x)$ in $(0, + \infty)$ is measurable? If yes, how can I calculate it?

Gage Potter 2022-04-23

How does this simplify to $\log \sqrt{x}$
How does $\log x - \log \sqrt{3} x - \log \sqrt{6} x$ simplify to $\log \sqrt{x}$
I've tried to get Bagatrix Algebra Solved! to solve it, but it even got the wrong answer... (I checked it by replacing x with 5 and typing it out on a calculator..)
No matter what I do, I end up with an answer that is correct and a bit simplified, but not as simplified as $\log \sqrt{x}$

Rylan Sullivan 2022-04-23

How to calculate the integral
$I = \int_{0}^{1} \frac{x^{n} - 1}{\ln (x)} d x$

Perla Galloway 2022-04-22

If z is complex variable, how to prove $lim_{z \to 0} \frac{z^{4}}{{| z |}^{2}} = 0$
I have simplified to $lim_{z \to 0} \frac{z^{4}}{z} \overset{―}{z} = lim_{z \to 0} \frac{z^{3}}{\overset{―}{z}}$
because ${| z |}^{2} = z \overset{―}{z}$
But how to progress?

kadetskihykw 2022-04-22

Solving a logarithmic equation
$\ln (x + 1) = 2 + \ln (x - 1)$ ; solve for x.
From there I get
$\ln \frac{x + 1}{x - 1} = 2 .$

estrigaslju 2022-04-22

Solve in terms of b:
$\log_{b} (1 - 3 x) = 3 + \log_{b} x$

yert21trey123z7 2022-04-22

Solve a simple equation with log in it
I'm stuck with solving this equation,
$2 \log x = \log 9$
This is how far I made it:
$\log x = \log 4, 5$
$x = ?$ I'm a beginner at logarithms so I appreciate ways to solve it and not just an answer.

Scientists measure a bacteria population and find that it is 2,700. Three days later, they find that the population has doubled. Write a function f(d) that could describe the bacteria population d days after the scientists first measured it, assuming it grows exponentially?

nrgiizr0ib6 2022-04-21

How to compute the asymptotic growth of $(\begin{matrix} n \\ \log n \end{matrix})$ ?

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