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
Given 3log5 xy2 + log5x = 4+2log5y, prove xy = 5.
Homework Equations
The Attempt at a Solution
Can i prove like this? Or i must use 3log5 xy2 + log5x = 4+2log5y to prove xy=5? Meaning my final result must be xy = 5? Please enlighten me..
Hello, I am having difficulty matching one term in my Laurent series to that which mathematica tells me is the correct answer. For the function
f(z)=log\frac{1+z}{1-z}
we know that there exists a k such that
Log|1+z|-Log|1-z|+i2\pi k
Now, we know that the Taylor series of f is as...
Edit: I answered my own question, I guess this thread serves no purpose so mods, you can delete this.
y = x^2
ln y = ln x^2
ln y = 2 ln x
Can we do the same thing with:
y = x^{2/x}
ln y = ln x^{2/x}
ln y = \frac{2}{x} x
Would that be correct? I just want to make sure because I used this...
Homework Statement
The relationship between arctanh and log is:
arctanh(x)=\frac{1}{2}log(\frac{1+x}{1-x})
but if i take x=1.5,
I have:
arctanh(1.5)=0.8047 + 1.5708i
and
\frac{1}{2}log(\frac{1+1.5}{1-1.5})=0.8047 + 1.5708i
as expected, but using the laws of logarithm, why does this...
Homework Statement
by substituting y = log2x solve for x in the following equation:
√log2x = logs2√x
Homework Equations
logab=c then a^c = b
The Attempt at a Solution
if y = log2x then the equation becomes √y = log2 x^1/2
this implies √y = 1/2 log2x which simplifies to √y = 1/2 y...
Homework Statement
i have some data and i have to find out if they can be put in the function f(x)=a^x or f(x)=x^a
it is about how much time there goes before the population soemwhere grows or such.
Homework Equations
data: it is about the growth or a population in a villlage (way hard to...
Homework Statement
Show that
\frac{1}{x} \ln \frac{x+1}{x-1} = \sum_0^\infty \frac{2x^{2n}}{2n+1}.
2. The attempt at a solution
I tried to use the relation
\frac{1}{1+x} = \frac{d}{dx}\ln (1+x)
and expand as a geometric series, but this did not lead anywhere since I then ended...
If I have log a b = c, what is the term for the value b? I know that a is the base and c is the exponent, but I can't think of what b is called! While I am at it, in ab = c, what exactly is c called? The power? Would that make b from the previous example the power as well? I have a...
Homework Statement
Consider a branch of \log{z} analytic in the domain created with the branch cut x=−y, x≥0. If, for this branch, \log{1}=-2\pi i, find the following.
\log{(\sqrt{3}+i)}
Homework Equations
\log{z} = \ln{r} + i(\theta + 2k\pi)
The Attempt at a Solution
This one...
We began learn flowcharts at school. First homework is make logarithm flowchart. I don't know how work logarithm cause we didn't learn in math yet. I tried make flowchart myself but I'm sure it's wrong cause don't know how logarithm works. Can someone help me?
Thanks
Hello, was wondering if anyone could please help me with the following questions as for math I have been given a substitute teacher who is of little help.
Any help would be much appreciated, even if its just pointing me in the right direction
Equation 1:
Log (x-3) = 1 + Log 4 - Log x...
Hello
I am currently studying Math B in high school and am having an extremely hard time with my current assessment. If anyone could please help me with the equations listed below it would be very much appreciated as my current Math teacher is on long service leave, and our substitute is...
Homework Statement
is ln(x)3 the same as saying [ln(x)]3? Also, if there is a difference, which one applies to the exponent being moved into the front of the logarithm as in 3ln(x)?
Homework Equations
natural logarithms
The Attempt at a Solution
I just got tripped up on the...
Homework Statement
The equation (attached as image) has
(a)one irrational solution
(b)no prime solution
(c)two real solutions
(d)one integral solution
i would like to get help on how to find the possible values of x
Homework Equations
( the equation is attached...
Homework Statement
Write the quantity using sums and differences of simpler logarithmic expressions. Express the answer so that logarithms of products, quotients, and powers do not appear.
log_{10}\frac{3}{\sqrt{1+x}}
Homework Equations
The Attempt at a Solution
The...
Homework Statement
2^{5x}=3^x(5^{x+3})
Homework Equations
The Attempt at a Solution
ln2^{5x}=ln3^x(ln5^{x+3})
5xln2=xln3(x+3)ln5
Here's where I get stuck... I've tried a bunch of different manipulation, but can't seem to isolate x...
Thanks in advance!
Homework Statement
y = 2^(sin x)
Find derivative.
Homework Equations
The Attempt at a Solution
y = 2^{sin x}
ln y = ln (2^{sin x})
ln y = sin x ln(2)
y' = [ln(2)] (cos x) (2^{sin x})
Yet, according to some rule in the book:
a^x derivative is ln a times a^x, or in...
I have a strange problem.
Have this example.
log(x-2)+log(x+1)=0
Domain of these functions are: x>2, x>-1 resulting in x>2;
by logarithm rule we can combine these 2 logarithms and make one logarithm.
log(x-2)(x+1)=0
(-infinity,-1) U (2,+infinity)
But the domain of this...
This is not a homework question. I just try it for enjoyment.
Let L = log to the base x of (yz) M = log to the base y of (xz) and
N = log to the base z of (xy)
This is how I do it without much luck.
I put all the equations in exponential form
yz = x^L xz = y^M xy...
Homework Statement
Write the expression as a logarithm of a single quantity
3log2-1/3log(x²-1)
Homework Equations
none
The Attempt at a Solution
3log2-1/3log[(x+1)(x-1)]
Homework Statement
Write the expression as a logarithm of a single quantity
3log5^a+4log5^b-2log5^c
Homework Equations
none
The Attempt at a Solution
??
Homework Statement
log2(x3)-3= 2log2x
Homework Equations
NoneThe Attempt at a Solution
Do I start by expanding the x to the power 3 to the log which then makes it 3log2x? or am I totally off track?
Any help is greatly appreciated. Thanks
Homework Statement
Find the integral of:
[a*ln(b/(b -cx)) - kx] dx
Where all a,b,c,k are constants and x is the variable.
Homework Equations
The Attempt at a Solution
Rewrote is:
a*INT(ln(b/(b-cx)) dx) - k*INT(x dx)
I don't know how to solve the first part, (the second...
I got this question and I know a bit on how to start it, but not sure which direction is best:
By considering the power series 1/(1 + x) and 1/(1 - x) show that:
so I do the differentiation - ln(1 + x) = x0 du/1 + u = x - x2/2 + x3/3 - x4/4 ...
which equals - = sigma (upper infinity and...
Simplify
3 log[5] (5^2)
By the way ... to enter logb(x) for some base b one types: log[b](x)
... but once you've simplified the given expression you won't need to know that!
3log[5] (5^2) = (3log(5))/(log(5))
therefore:
3 log[5] (5^2) = 6
Although, would i keep the...
Homework Statement
Prove that the following is true:
n
\Sigma lg k = \Theta (n lgn)
k=1
Homework Equations
the lg in this case is base 2
The Attempt at a Solution
i don't kno how to apply geometric or arithmetic progression to the 1st part
i was trying to substitute for k but that wasnt...
Homework Statement
The question is located here http://i51.tinypic.com/2cge9mt.jpg
Homework Equations
My a value is -3
my b value is -3sqrt(2)
my c value is -2.4
The Attempt at a Solution
1) ln(-3 - 3 sqrt(2) i)
= Ln |-3 - 3 sqrt(2) i| + i arg(-3 - 3 sqrt(2) i)
= ln...
Homework Statement
If log4 N=p and log12 N=q, show that
log3 48=
Homework Equations
The Attempt at a Solution
I tried by substituting p and q into but i couldn't get the required answer. Can anyone help?
I am not very good in logs i have tried to attempt below problem in vain..give me an insight.
Prove Log(√27 + Log √8 - Log √125 ) / (Log 6 – Log 5) =3/2
I am not good in logarithms how do you show this…
This is how I did it:,
Log ( ( 3√3 *2√2 ) / 5√5 ) ) / ( 6/5)
((3√(3...
Homework Statement
find -Ln(1-e(i\theta) (in terms of theta)
(this is me just skipping the part of the problem I know and going straight to what I can't figure out)
Homework Equations
ln(z) = ln(rei\theta)=Ln(r) + ln(ei\theta) = Ln(r) + i\theta
The Attempt at a Solution
I don't really...
May I ask how to find the E\left(ln x\right) and Var\left(ln x)?
The X_{i} are random sample from the f\left(x; \theta\right) = \theta x^{\theta - 1}I_{\left(0, 1\right)}\left(x\right) where \theta > 0.
I need the information in finally solving the Cramer-Rao lower bound for the variance of...
Why is there a logarithm in the entropy formula? Why is it S=kln(N) where k is the Boltzmann constant and N is the number of microstates? Why isn't it S=N?
I've been asked to express the inverse hyperbolic secant function arcsech in terms of the natural logarithm and am unsure as to where to begin in solving such a problem?
could someone please point me in the right direction?
How do I start to evaluate this integral?
\int\frac{1}{\ln x}-\frac{1}{(\ln x)^2} dx
I tried subbing u=\ln x but I'm getting no where...
The answer is
[tex]\frac{x}{\ln x}+C[/itex]
If I differentiate the answer, I get the integral easily, but the reverse... I'm having trouble figuring out...
Homework Statement
y = log base 3 e^2x
Homework Equations
The Attempt at a Solution
I got y' = 2e^2x divided by e^2xln3
is this right?
Sorry for the pathetic way of presenting this. I haven't been able to use the lancet program for proper ways to write mathematical stuff
Homework Statement
y=ae^x
Homework Equations
rearrange to find a
The Attempt at a Solution
y/a=e^x
x=ln(y/a)
x=lny-lna
lny-x=lna
now how do I rearrange/inverse to seclude a
How do you differentiate :
2^n/2?
You can't you just use the power rule?
The correct answer is 2^n/2 (In^2) 1/2
lg^2 x n
Where lg is log base 2.
The correct answer is 2lgn 1/n^2 . 1/n
Why is this so? Isn't lg^2 = 1? And differentiate n and we get 1?
Sorry if I sound...
Homework Statement
Simplify x^(3logx2 - logx5) to find an exact numerical value.
Homework Equations
The Attempt at a Solution
3logx2=logx2^3 or logx8,
(logx8 - logx5)=logx8/5
the inverse would be x^y=8/5 (y is unknown)
therefore logx8/5=y and x^(logx8/5)=x^y=8/5
and the...
I'm very new to logarithm.
I've got homework to solve a ques using logarithm i.e multipication of 9356 by 0.396. And I've done it in following manner
9356*0.396
=log 9356 + log 0.396
=3.9711-1.4023
=2.5688
I know I've done it in a wrong manner as the ans is wrong. I'm very new to it. I...
Homework Statement
If X is a random variable with density function: f(x) = \lambdae^{-x \lambda}where X>=0.Homework Equations
Why is the expected value of X, or E[X] = \frac{1}{\lambda}?The Attempt at a Solution
E[X] = \int x*(\lambdae^{- \lambda}^{x}) dx, where the integral is from 0 to...
Homework Statement
Prove: \sum_{i=1}^{\infty}\sum_{j=1}^{i-1} \frac{(-1)^i}{i j}=\frac{1}{2}\ln^2 2
Homework Equations
\ln 2 = \sum_{i=1}^{\infty} \frac{(-1)^{i+1}}{i}
The Attempt at a Solution
I'm trying to manipulate the l.h.s. of the problem to transform it into...
Homework Statement
I need to prove that the following limit holds when
\sqrt{s}>>m
\log\left(\frac{\sqrt{s}+\sqrt{s-4m^2}}{\sqrt{s}-\sqrt{s-4m^2}}\right) \rightarrow \log\left(\frac{s}{m^2}\right)
The Attempt at a Solution
I've tried several manipulations using logarithms properties but had...
If we define a function f(x) such that:
f(x) = \int_{1}^x \frac{dt}{t}
for x>0, so that:
f(y) = \int_{1}^y \frac{dt}{t}
and
f(xy) = \int_{1}^{xy} \frac{dt}{t}
is there a way, using just these "integral" definitions, to prove that:
f(x) + f(y) = f(xy)
Clearly, the function...
Homework Statement
the inverse laplace transform of ln\frac{s+2}{s-5} using the inverse Laplace transform of the derivative
Homework EquationsL^{-1}{\frac{d^{n}}{ds^{n}}F(S)} = (-1)^{n}t^{n}f(t)
The Attempt at a Solution
the integral of ln\frac{s+2}{s-5} I worked to be (s+2)ln(s+2)-(s+2)...
Homework Statement
Hi all, I'm having some trouble seeing why this question isn't trivial, maybe someone can help explain what I actually need to show - shouldn't take you long! :)
Suppose h:\mathbb{C} \to \mathbb{C}-\{0\} is analytic with no zeros. Show there is an analytic function...
Hi all,
Do you know how to solve the following inequations?
\ln \left( {\frac{{x + a}} {{x + b}}} \right) \leq cx + d
\ln \left( {\frac{{x + a}} {{x + b}}} \right) \geq \frac{{x^3 + cx^2 + dx + e}} {{ux^2 + vx}}
a, b, c, d, e, u, v are constants.
x is a variable.
Can you suggest...