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.
Hello again!
I have been working on this log, and the longer I work on it, the more confused I get! Here's the problem:
Find the exact value for:
ln(ln[e^{e^{5}}])
----
Here's what I've tried so far:
e(ln[e^{e^5}}])
e^{x} = ln(e^{e^5}})
e^{x} = e^{e^5}}
e^{5} = (2.72)^{5}...
[SOLVED] Logarithm overkill!
Hello again!
I have been working on this log, and the longer I work on it, the more confused I get! Here's the problem:
Find the exact value for:
ln(ln[e^{e^{5}}])
----
Here's what I've tried so far:
e(ln[e^{e^5}}])
e^{x} = ln(e^{e^5}})
e^{x} =...
Does anyone know if the property of the logarithm function that:
log(ab)=log(a)+log(b)
is unique to that function? In other words, has it been shown that there can be no other function with that property?
-j
Well I came across this when someone asked me this question:
(-2)^n = 16
I can clearly see n=4. However, he did this:
ln((-2)^n) = ln(16)
n*ln(-2) = ln(16)
n*ln(2)+n*i*pi = ln(16)
How can I show that n=4 from this?
Hi everyone.
I know this question is quite simple but I can't wrap my head around it at the moment..
Solve for y: ln(x) + ln(y) = 0
I've tried differentiating both terms and then arranging for y but I get y = -x. The answer is meant to be y = 1/x.
Thanks all!
Hi,
Okay, I just went over some old stuff this week to prepare for the next vast vast, upcoming test.
I came across a Logarithmic Equation book, well, it's more than 300 pages long, it covers most things from Exponential to Logarithm. :woot: I have read about 1/4 of it, and I was stuck...
Homework Statement
Given that
\log_{4n} 40\sqrt {3} = \log_{3n} 45
find
n^3
Homework Equations
Logarithm properties
The Attempt at a Solution
I can get an expression for n but looks messy, and suspect there is probably a more compact answer. This is what I did...
Homework Statement
xln(2x+1)-x+\frac{1}{2}ln(2x+1) = \frac{1}{2}(2x+1)ln(2x+1)-xHomework Equations
ln(x^a) = aln(x), ln(xy) = ln(x) + ln(y), ln(\frac{x}{y}) = ln(x) - ln(y)The Attempt at a Solution
I have no idea how you can go from xln(2x+1)-x+\frac{1}{2}ln(2x+1) to...
Actually, I am trying to use what I have learned on school to something else
Homework Statement
3*10^x = 1.73*10^14
Homework Equations
lga^x = xlga
The Attempt at a Solution
10^x = (1.73 * 10^14)/3
10^x = 5.767 * 10^13
xlg10 = 13lg(57.67)
x=13lg(57.67)
x=22.89
:\...
Homework Statement
Show that
ln(z^\alpha) = \alpha ln(z)
where 'z' and 'alpha' are complex.
Homework Equations
ln \alpha = ln r + i(\theta + 2*n*\pi)
The Attempt at a Solution
For the left hand side I have ln (z^\alpha) = ln...
Hey!
I have always learned that functions like logarithms, exponentials, trigonometrics etc. have to operatore on pure numbers and not numbers with units. For instance, you cannot write:
Sin ( 5 kg*m/s^2 )
But in chemistry I often find formulas where logarithmes of numbers with units...
Hi all,
I'm a bit puzzled by one of my homework questions. I got an answer, but I have nothing to check it with and I'm not sure that my answer is correct.
The question states that y=ln(ln(ln(x))), and asks for y'. This is what I've done, but it seems a bit too simple to me...
I was doing some assignment i have to give in, for math, and came upon this exponential equation: (2^x+1) + (2^x+2) = (2^1-x) + (2^3-x)
I thought, pfft, that's easy...so i did it, wrong answer, tried something else, wrong answer..tried another tactic, and i think you can guess what...
I need help defining a logarithm.
My book simply says: A logarithm is an exponent.
This stumped me because I can't see how that is. I don't know what question to ask, but I might not be apprehending the relationship between an expo. function and a log. function.
#2
Hi,
Well can anyone tell me how to find the natural logarithm of a complex number p + iq.
Also please tell me how to convert it into logarithm to the base 10.
An external link to a webpage (where all the details are given) will be appreciated.
Confused, but tried it this way:
Use u-substitution to show that (for y a positive number and x>0)
\int_{x}^{xy} \frac{1}{t} dt = \int_{1}^{y} \frac{1}{t} dt
so, u=t and du=dt
if x=1
t=xy u=y(1)=y
t=x u=1
or
u=1/t and du/ln [t] = dt
if x=1
t=xy u=1/y
t=x y=1
Thanks for...
The question is this:
Consider p(z) a polynomial and C a closed path containing all the zeroes of p in its interior. Compute
\frac{1}{2\pi i}\int_C z\frac{p'(z)}{p(z)}dz
The solution given by the manual starts by saying that
\frac{p'(z)}{p(z)}=(log(p(z)))'.
But there is no...
my physics teacher told us that for now well be solving the equation of a line such as y=kx^n by trial and error for finding "n" using the multiplicative change in 2 points i.e x_1,y_1,x_2,y_2.
what we would do is for example 2 points (13,1) (6, 1.4):
n=\frac{y_2}{y_1}=(\frac{x_2}{x_1})^n...
Hello sorry to post another so soon. This book is VERY hard to learn from... its not very good at explaining. The question says:
Evaluate the following logarithm: 10log1019. It never showed me how to do it when there's a number in front of the log. What I got from the log 1019 is...
hi
could anyone tell me where I went wrong ?
simultaneously solve
2logbase2 y = logbase4 3 + logbase2 x
3^y = 9^x
But for the top I get y = 3 root x
and bottom I get y=3x
so what's gone wrong ?
thanks
roger
I'm puzzled by...
\ln \left( -\frac{1}{2} \right) = - \ln \left( 2 \right) + i \pi
Why is this true? How can I possibly get this result?
I know that
\ln \left( \frac{1}{2} \right) = - \ln \left( 2 \right).
Thank you so much
My book have a really crappy proof of how
log a^x = x log a
can be true. Can someone help me?
Another question deals with an application of it which made me really confused:
Factorise (4a^3) - (29 a^2) + 47a - 10
then solve (4*4^3x) - (29*4^2x) + 47*4^x - 10 = 0
I see...
Hey
I am doing an investigation for logarithms, and I have a question. logx^n = nlogx. Based on previous knowledge of exponents, could someone please explain why this is true? Thanks.
I'm trying to use LOGs other than log base 10 and base e on my TI-86. Can I accomplish this like this?:
log base a of b = (log base 10 of b) / (log base 10 of a) ?
or is it:
log base a of b = (ln b) / (ln a) ?
Help needed ASAP. Thanks!
I've just introduced myself to logarithms and have done most of the questions, but am having trouble with one or two of them:
Q1: Find values of x for which:
Log(to base 3)x - 2log(to base x)3 = 1.
I have no idea where to start on this question.
Q2: Solve:
25^x = 5^(x+1) -6.
On...
I have a small problem with logarithms. We have to solve physics and chemistry problems using only logs. And I don't know how to do the following -
1.2341 + bar2.4412
Well actually my question is how do you add or subtract bar numbers ? (It's the numbers with '-' on top)
Rohit.
What I have not seen in books about the Gibbs paradox is that it doesn't exist if we make the Gibbs correction at the logarithm of the Z function, not at the Z function itself, in that way:
\ln Z_{i} - \ln N_{i} !
where N_{i} is the number of identical particles of class i, where there...
can some one explain to me how is taking the logarithm of euler product gives you -sum(p)[log(1-p^s)]+log(s-1)=log[(s-1)z(s)]?
my question is coming after encoutering this equation in this text in page number 2...
Their must be over a million definitions involving the constant "e". What I would like is a description of the natural logarithm in natural terms, not just saying "e is where
e
[inte] dx/x=1, etc."
1
In other words, why choose this function to define e, and how does it most...