Questions tagged [logarithms]
Questions related to real and complex logarithms.
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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?
$x^2+2x-8 = 16$
$x^2+2x = 24$
$...
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Name for logarithm variation that works on non-positive values?
I've come up with the following variation of a logarithm, intended to work on values that can be 0, or can grow exponentially from zero in either positive or negative direction.
$$myLog(x) =
\begin{...
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prove $\log_{10} (2)$ is irrational [duplicate]
Possible Duplicate:
About irrational logarithms
Please help proving that $\log_{10}(2)$ is irrational.
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Understanding imaginary exponents
I am trying to understand what it means to have an imaginary number in an exponent. What does $x^{i}$ where $x$ is real mean?
I've read a few pages on this issue, and they all seem to boil down to ...
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Bounding the series $\sum_{m\leq x,m\neq n}\frac{1}{|\log(m/n)|}$
I am trying to reproduce the following bound:
$\sum_{1\leq m\leq x, m\neq n}\frac{1}{|\log(m/n)|}=O(x\log(x))$,
for $x\geq 2$ and some $n$, $1\leq n\leq x$ (the implicit constant shouldn't depend on ...
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About irrational logarithms
Could someone provide, please, a proof of the theorem below?
"Being $x$ and $b$ integers greater than $1$, which can not be represented as powers of the same basis (positive integer) and integer ...
- 4,260
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Why does an equiangular spiral become logarithmic (intuitively)?
One of the most famous 2D-curves are logarithmic spirals (or Spira mirabilis). They can be constructed by using a machinery that ensures a constant angle between the tangent and the radial lines all ...
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Is it true that $x^{\log_z(y)} = y^{\log_z(x)}$?
it has been years since I have done logs, I remember something like this:
$$x^{\log_z(y)} = y^{\log_z(x)}$$
(where $z$ is the base)
Is that correct? It doesn't seem so, since
$$3^{\log_2(4)} \neq ...
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What is the $x$ in $\log_b x$ called?
In $b^a = x$, $b$ is the base, a is the exponent and $x$ is the result of the operation. But in its logarithm counterpart, $\log_{b}(x) = a$, $b$ is still the base, and $a$ is now the result. What is $...
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How did the notation "ln" for "log base e" become so pervasive?
Wikipedia sez:
The natural logarithm of $x$ is often written "$\ln(x)$", instead of $\log_e(x)$ especially in disciplines where it isn't written "$\log(x)$". However, some mathematicians disapprove ...
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What's so "natural" about the base of natural logarithms?
There are so many available bases. Why is the strange number $e$ preferred over all else?
Of course one could integrate $\frac{1}x$ and see this. But is there more to the story?
user218
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Exponential and log functions compose to identity
How to prove that the exponential function and the logarithm function are the inverses of each other? I want it the following way. We must use the definition as power series, and must verify that all ...
user218
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Sum of the alternating harmonic series $\sum_{k=1}^{\infty}\frac{(-1)^{k+1}}{k} = \frac{1}{1} - \frac{1}{2} + \cdots $
I know that the harmonic series $$\sum_{k=1}^{\infty}\frac{1}{k} = \frac{1}{1} + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{5} + \frac{1}{6} + \cdots + \frac{1}{n} + \cdots \tag{I}$$ diverges,...
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