In mathematics, the logarithm is the inverse function to exponentiation. That means the logarithm of a given number x is the exponent to which another fixed number, the base b, must be raised, to produce that number x. In the simplest case, the logarithm counts the number of occurrences of the same factor in repeated multiplication; e.g., since 1000 = 10 × 10 × 10 = 103, the "logarithm base 10" of 1000 is 3, or log10(1000) = 3. The logarithm of x to base b is denoted as logb(x), or without parentheses, logb x, or even without the explicit base, log x, when no confusion is possible, or when the base does not matter such as in big O notation.
More generally, exponentiation allows any positive real number as base to be raised to any real power, always producing a positive result, so logb(x) for any two positive real numbers b and x, where b is not equal to 1, is always a unique real number y. More explicitly, the defining relation between exponentiation and logarithm is:
log
b
(
x
)
=
y
{\displaystyle \log _{b}(x)=y\ }
exactly if
b
y
=
x
{\displaystyle \ b^{y}=x\ }
and
x
>
0
{\displaystyle \ x>0}
and
b
>
0
{\displaystyle \ b>0}
and
b
≠
1
{\displaystyle \ b\neq 1}
.For example, log2 64 = 6, as 26 = 64.
The logarithm base 10 (that is b = 10) is called the decimal or common logarithm and is commonly used in science and engineering. The natural logarithm has the number e (that is b ≈ 2.718) as its base; its use is widespread in mathematics and physics, because of its simpler integral and derivative. The binary logarithm uses base 2 (that is b = 2) and is frequently used in computer science. Logarithms are examples of concave functions.Logarithms were introduced by John Napier in 1614 as a means of simplifying calculations. They were rapidly adopted by navigators, scientists, engineers, surveyors and others to perform high-accuracy computations more easily. Using logarithm tables, tedious multi-digit multiplication steps can be replaced by table look-ups and simpler addition. This is possible because of the fact—important in its own right—that the logarithm of a product is the sum of the logarithms of the factors:
log
b
(
x
y
)
=
log
b
x
+
log
b
y
,
{\displaystyle \log _{b}(xy)=\log _{b}x+\log _{b}y,\,}
provided that b, x and y are all positive and b ≠ 1. The slide rule, also based on logarithms, allows quick calculations without tables, but at lower precision.
The present-day notion of logarithms comes from Leonhard Euler, who connected them to the exponential function in the 18th century, and who also introduced the letter e as the base of natural logarithms.Logarithmic scales reduce wide-ranging quantities to tiny scopes. For example, the decibel (dB) is a unit used to express ratio as logarithms, mostly for signal power and amplitude (of which sound pressure is a common example). In chemistry, pH is a logarithmic measure for the acidity of an aqueous solution. Logarithms are commonplace in scientific formulae, and in measurements of the complexity of algorithms and of geometric objects called fractals. They help to describe frequency ratios of musical intervals, appear in formulas counting prime numbers or approximating factorials, inform some models in psychophysics, and can aid in forensic accounting.
In the same way as the logarithm reverses exponentiation, the complex logarithm is the inverse function of the exponential function, whether applied to real numbers or complex numbers. The modular discrete logarithm is another variant; it has uses in public-key cryptography.
Homework Statement
Verify the formula logbx = logax/logab for a, b > 0
Homework Equations
The Attempt at a Solution
I don't even know how I would go about starting this...
How many integers will satisfy x in the inequality:
1< \log_{3}({\log_{2}{x})}< 2
Note: The log there is not multitplied to the other log. The log there I think is read like this, logarithm of logarithm of x to the base 2 to the base 3.
What can be the solution or technique for this...
Hello,
I made a mistake in the title of this thread and this question is on general logarithms;
loga(aloga(x)) = loga(x) ==> aloga(x) = x
Can someone enlighten me on why loga(aloga(x)) simplifies to loga(x)? Can someone prove why this is true? Futhermore, why does this imply that...
This equation specifies displacement x in terms of motion time t (starting from rest).
x = \tau g\left(t + \tau e^{-t/\tau} -\tau\right)
where tau = m/b is the system time constant of a mass m suspended from a mechanical braking device with rectilinear damping constant b and g is standard...
How does
(ln(x))^(1/x)=ln(x^(1/x))?
A friend told me this was a true statement but could'nt prove it. If that isn't true, then how would you find the lim x->0 of (ln(x))^(1/x) using L'Hospital's Rule?
Homework Statement
Positive integers a and b, where a < b, satisfy the equation :
ab = ba
By first taking logarithms, show that there is only one value of a and b that satisfies the equation and find the value !
Homework Equations
logarithm
The Attempt at a Solution
I know the...
Homework Statement
e^z = e
e^z = e^-z
Homework Equations
they are the first 2 questions on my homework and my book is so bad i don't even know how to get the right answers, I've been online for over an hour I'm i'm sure they are easy.
The Attempt at a Solution
ffirst one just...
I read the following expression in a book:
\int_{-\infty}^{\infty} \dfrac{1}{t(1-t)} \log \left| \dfrac{t^{2} q^{2}}{(p-tq)^{2}} \right| ~ dt = - \pi^{2}
p and q are both timelike four-vectors, so p², q² > 0
This integral was solved by using the identity
\lim_{s \to \infty}...
First of all, greetings to the scientific community here at Physics Forums.
The following set of equations is given:
y^(x-3y) = x^2
x^(x-3y) = y^8
With the next assumption given: x-3y is unequal to (-4).
My attempt was to isolate the y variable in both of the equations so that i will...
Homework Statement
Hi everyone
I need help for this problem :
If 2*\log_2 (x-2y)=\log_3 (xy) , find \frac{x}{y}
Homework Equations
\log_bx = \frac{\log_ax}{\log_ab}
The Attempt at a Solution
2*\log_2 (x-2y)=\log_3 (xy)
\log_2 (x-2y)^2=\log_3 (xy)
\frac{\log_2...
does anyone know how to calculate (in the sense of distribution) the Fourier transform of
f(x)= ln|x|
that is to obtain the integral \int_{-\infty}^{\infty} dx ln|x|exp(iux)
Homework Statement
Hey all, I just need some help remembering my basic simplification rules.
Can the expression eC*ln(x) be simplified anymore. It would certainly help out an integral I'm working on but I don't remember my simplification real well.
Homework Equations
elnx=xThe Attempt at a...
Suppose X\in\mathbb{R}^{n\times n} is orthogonal. How do you perform the computation of series
\log(X) = (X-1) - \frac{1}{2}(X-1)^2 + \frac{1}{3}(X-1)^3 - \cdots
Elements of individual terms are
((X-1)^n)_{ij} = (-1)^n\delta_{ij} \;+\; n(-1)^{n-1}X_{ij} \;+\; \sum_{k=2}^{n} (-1)^{n-k}...
I am trying to explore a number of things regarding the entropy of random strings and am wondering how a character set of random size would affect the entropy of strings made from that set.
Using the following formula, I need to take the log of a discrete random variable
H = L\log_2 N...
Homework Statement
1) Solve the equation using the exponential form of the equation.
Log2(2-5x)=7
2) Rewrite as a single logarithmic expression:
3log x +(1/2)log z
Homework Equations
The Attempt at a Solution
1) I have no idea how to solve this.
2) I know that two logarithms multiply to...
I know that there is a year old thread that has just been resurrected that is slightly on this topic, but I had a question that I thought might merit a new thread.
I'm working with my Calculus book here, and I'm working on the chapter called, "Logarithmic Functions from the Integral Point of...
Homework Statement
Let I be the identity matrix, and let N be a nilpotent matrix order k. Then the matrix exponential function is defined as:
exp(N) := I + N + (1/2!)*N^2 + (1/3!)*N^3 + ... + (1/k!)*N^k
Similarly, we may define the matrix logarithm as the function
-log(N) := I - N +...
hello,
I was trying to figure out what will be the y value for this equations:
1 - e^-0.15*10^-5*y = 0.1
Could somebody help me in this?? The answer is supposed to be 70,240.
thanks.
Hi!
Does anyone know what a branch-cut singularity is? I have been trying to understand its importance in physics, but I got lost. I would guess that a singularity in physical context should mean that the value of a function should become very big near that singularity. But if we take complex...
Homework Statement
log2(x+2)=log2x2
-log base 2(X+2)= log base 2 (x2)
The Attempt at a Solution
I know the answer is supposed to be -1 and 2, but I get the wrong answer every time I try.
I tried bringing log2x2 over to the other side and then got log2(2/x) which got me nowhere...
Homework Statement
Rewrite the expression as a sum, difference or multiple of logs.
log55x2
Homework Equations
The Attempt at a Solution
log55x2 =
2log55x =
...
Homework Statement
\displaystyle\int\left(\dfrac{\ln x}{x}\right)^2 dx
2. The attempt at a solution
I tried letting u=\ln^2 x and dv the rest and I also tried dv=\ln^2 x dx and u the rest. It won't work out.
Homework Statement
Hi.
I want to solve:
\frac{d\ln(-x)}{dx}.
When using the chain rule I get:
\frac{d \ln(-x)}{dx} = \frac{d\ln(-x)}{d(-x)}\frac{d(-x)}{dx} = -\frac{d\ln(-x)}{d(-x)}.
But how do I find the last derivative? I know by experience that it is -x-1, but how is the derivation...
I have been working on a solid state physics homework problem, and I have gotten the answer down to an integral that I am unsure how to do by hand. I can plug it into Mathematica, and I receive the correct answer (I am asked to show something) but I would like to know how to do this integral by...
Homework Statement
This one has has me pretty lost, I'm not at all sure what to do, I've tried a couple of things. I have to find x.
2(x-1) = 5(73x)
Homework Equations
Logarithm laws?
The Attempt at a Solution
(did multiple attempts but my most reasonable to my account)
2(x-1) =...
Homework Statement
Find x in the following equation
(ln 64 / 2) + ln x = ln 18 - ln x
The answer should be x = 3/2
Homework Equations
Logarithm laws
ln m - ln n = ln (m/n)
The Attempt at a Solution
I tried two methods:
(ln 64 / 2) + ln x = ln 18 - ln x
ln 32 + ln x = ln 18...
Hello,
I understand the current concept of the characteristic however there either is a mistake in my answerbook or I made a mistake
log15 850 = 850(1/15) = 1.568
The characteristic should be 1 and the mantissa is 0.568 -- My answer book says it's 2. I have done all my other exercices...
Homework Statement
It was found that the percentage of the carbon-14,P, contained in the bones of an animal n years after it has died is given by P = 2^{-kn} , where k is a positive constant. The percentage of carbon-14 contained in the bones after the animal has been dead for 5668 years was...
Homework Statement
Prove:
1/log3a + 1/log4a = 1/log12a
Homework Equations
ay=x
Logarithms rules (addition, subtraction, power, etc.)
logax=logbx/logba
The Attempt at a SolutionLeft Side:
1/log3a + 1/log4a
=log3a+log4a/log12a (via common denominator)
The problem is how to add logarithms...
Homework Statement
Express 2log_{2}X=1+log_{a}(7X-10a) find x in terms of a.
i wondering if there is other methods to solve aside from completing the sq.
Homework Equations
The Attempt at a Solution
Homework Statement
Homework Equations
I got x²-7ax+10a²=0...
Homework Statement
Write expression as a logarithm of a single quantity and then simplify if possible.
(1/4)[log (x²-1)-log (x+1)]+3log x
Homework Equations
The Attempt at a Solution
I got (1/4)log(x-1) + 3log x so far
Hello!
Run into trouble...again.
This concerns the inverse function of a logarithm
If a function maps x on to logax, then the inverse maps logax on to x.
So, f(x) = logax, can be presented as y=logax; therefore, x=ay.
The book states that the inverse is ax, why is this the inverse?
I tried...
I am not a physics or math student ,but i am interested in physics i want to understand the nature So i started studying physics my main source is internet .So i need help from people like U .
1.what is limits how and where it is used ?
2.what is logarithm how that table is made?
3.what...
I'm taking a complex variables course, and I'm really stuck at it, I've never felt this way in any math course before :S, I'm starting to get angry. Anyway here is the problem, I hope someone can give me a hand. I believe this is a basic and simple problem in the subject...
Homework Statement...
I was wondering if the Golden ratio base (phinary system) has any use somewhere and if arithmetics with it is easy?
I programmed a surprisingly simple algorithm to calculate the logarithm yielding digits in base phi using nothing more than 2 multiplications/divisions per result digit. Can it...
I was recently doing some problems with Entropy, and came to the startling conclusion of incorrect units for most of my answers!
One problem, for example--was to calculate the change in entropy of a given sample as it gained heat per temperature.
\Delta S=\int^{f}_{i}\frac{dQ_{r}}{T}
The...
Hello
So I have a problem, which is to use integration by parts to integrate...
\int^{1}_{0}(1-x) ln (1-x) dx
The way I have been working is it to separate it out into just...
\int^{1}_{0}ln (1-x) dx - \int^{1}_{0}x ln (1-x) dx
and then integrating by parts on each of these...