Questions tagged [logarithms]

Questions related to real and complex logarithms.

The logarithm is generally defined to be an inverse function for the exponential. If $x > 0$ is a real number and $b > 0$, $b \ne 1$, then the base-$b$ logarithm is defined by

$$\log_b(x) = y \iff b^y = x$$

The most commonly used bases are base $10$ and $2$ (which frequently arises in computer science), and particularly base $e$. The natural logarithm $\ln$ is defined to be $\log_e$.

Alternatively, the natural logarithm can be defined to be a primitive of the function $$f(t) = \frac{1}{t}$$ subject to the condition that $\ln{1} = 0$.

In the study of complex numbers, the solutions $a$ of $e^{a} = z$ are called complex logarithms. This uniquely specifies the modulus of $a$, but not its argument; as such, we define the principal logarithm $\operatorname{Log}(re^{i\theta}) = \ln{r} + i \theta$, with the restriction $-\pi < \theta \le \pi$ (or alternatively, $0 \le \theta < 2\pi$). This leads to a branch cut, or discontinuity - alternatively, the complex logarithm can be viewed as a multi-valued function.

Reference: Logarithm.

This tag often goes along with .

9332 questions
16 answers

Alternative notation for exponents, logs and roots?

If we have $$ x^y = z $$ then we know that $$ \sqrt[y]{z} = x $$ and $$ \log_x{z} = y .$$ As a visually-oriented person I have often been dismayed that the symbols for these three operators look nothing like one another, even though they all…
  • 2,525
  • 4
  • 21
  • 18
27 answers

Unexpected examples of natural logarithm

Quite often, mathematics students become surprised by the fact that for a mathematician, the term “logarithm” and the expression $\log$ nearly always mean natural logarithm instead of the common logarithm. Because of that, I have been gathering…
José Carlos Santos
  • 397,636
  • 215
  • 245
  • 423
10 answers

The deep reason why $\int \frac{1}{x}\operatorname{d}x$ is a transcendental function ($\log$)

In general, the indefinite integral of $x^n$ has power $n+1$. This is the standard power rule. Why does it "break" for $n=-1$? In other words, the derivative rule $$\frac{d}{dx} x^{n} = nx^{n-1}$$ fails to hold for $n=0$. Is there some deep…
7 answers

Prove $\left(\frac{2}{5}\right)^{\frac{2}{5}}<\ln{2}$

Inadvertently, I find this interesting inequality. But this problem have nice solution? prove that $$\ln{2}>(\dfrac{2}{5})^{\frac{2}{5}}$$ This problem have nice solution? Thank you. ago,I find…
  • 1
  • 15
  • 119
  • 475
21 answers

How do you explain the concept of logarithm to a five year old?

Okay, I understand that it cannot be explained to a 5 year old. But how do you explain the logarithm to primary school students?
  • 1,255
  • 1
  • 10
  • 11
11 answers

Demystify integration of $\int \frac{1}{x} \mathrm dx$

I've learned in my analysis class, that $$ \int \frac{1}{x} \mathrm dx = \ln(x). $$ I can live with that, and it's what I use when solving equations like that. But how can I solve this, without knowing that beforehand. Assuming the standard rule…
  • 991
  • 1
  • 9
  • 11
6 answers

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?
2 answers

Ramanujan log-trigonometric integrals

I discovered the following conjectured identity numerically while studying a family of related integrals. Let's set $$ R^{+}:= \frac{2}{\pi}\int_{0}^{\pi/2}\sqrt[\normalsize{8}]{x^2 + \ln^2\!\cos x} \sqrt{ \frac{1}{2}+\frac{1}{2}\sqrt{ \frac{1}{2}+…
Olivier Oloa
  • 118,697
  • 18
  • 195
  • 315
3 answers

What algorithm is used by computers to calculate logarithms?

I would like to know how logarithms are calculated by computers. The GNU C library, for example, uses a call to the fyl2x() assembler instruction, which means that logarithms are calculated directly from the hardware. So the question is: what…
  • 4,158
  • 4
  • 26
  • 42
5 answers

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 allow solving for logs with negative bases or with…
3 answers

Closed form for $\int_0^1\log\log\left(\frac{1}{x}+\sqrt{\frac{1}{x^2}-1}\right)\mathrm dx$

Please help me to find a closed form for the following integral: $$\int_0^1\log\left(\log\left(\frac{1}{x}+\sqrt{\frac{1}{x^2}-1}\right)\right)\,{\mathrm d}x.$$ I was told it could be calculated in a closed form.
4 answers

Integrals of the form ${\large\int}_0^\infty\operatorname{arccot}(x)\cdot\operatorname{arccot}(a\,x)\cdot\operatorname{arccot}(b\,x)\ dx$

I'm interested in integrals of the form $$I(a,b)=\int_0^\infty\operatorname{arccot}(x)\cdot\operatorname{arccot}(a\,x)\cdot\operatorname{arccot}(b\,x)\ dx,\color{#808080}{\text{ for }a>0,\,b>0}\tag1$$ It's…
4 answers

Integral $\int_0^1\frac{\ln\left(x+\sqrt2\right)}{\sqrt{2-x}\,\sqrt{1-x}\,\sqrt{\vphantom{1}x}}\mathrm dx$

Is there a closed form for the integral $$\int_0^1\frac{\ln\left(x+\sqrt2\right)}{\sqrt{2-x}\,\sqrt{1-x}\,\sqrt{\vphantom{1}x}}\mathrm dx.$$ I do not have a strong reason to be sure it exists, but I would be very interested to see an approach to…
1 answer

Proving that $\int_0^\pi\frac{x\ln(1-\sin x)}{\sin x}dx=3\int_0^\frac{\pi}{2}\frac{x\ln(1-\sin x)}{\sin x}dx$

Prove without evaluating the integrals that:$$2\int_0^\frac{\pi}{2}\frac{x\ln(1-\sin x)}{\sin x}dx=\int_\frac{\pi}{2}^\pi\frac{x\ln(1-\sin x)}{\sin x}dx\label{*}\tag{*}$$ Or equivalently: $$\boxed{\int_0^\pi\frac{x\ln(1-\sin x)}{\sin…
  • 23,476
  • 2
  • 61
  • 142
12 answers

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, but what about the alternating harmonic series…
2 3
99 100