Logarithms Definition and 257 Threads

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.

View More On Wikipedia.org
  1. Z

    Evaluating logarithms given log(a)b = 3

    Hi there Can someone please review my work and advise. Homework Statement Given log(a)b=3 , use the properties of logs to evaluate 4log(a)(1/b)+2log(a)b^3 Homework Equations I used the rules log(a)1 = 0 log(a)(x/y) = log(a)x-log(a)y log(a)x^m = mlog(a)x The...
  2. G

    Measuring the Richter Scale with logarithms

    An Earthquake measures 6.1 on the Richter Scale. What is the rating on an earthquake that is twice as powerful? I = I0 × 10M I-intensity of earthquake I0-earthquake with intensity of 0 M-magnitude of earthquake on the Richter Scale So I figured I'd use I1 = I0 × 106.1 to find the...
  3. S

    Logarithms/intro to logarithms (how did they the get solution?)

    Homework Statement Homework Equations The Attempt at a Solution I have no idea how they came up with 7/2 as a solution...can anyone tell me how? I am not sure how logarithms and square roots work...how did they get the 1/2 exponent over the second 2? What is the...
  4. K

    Predicting number of digits with logarithms

    If log103 =0.477 then the number of digits in 340 will be? 100.477 = 3 10(0.477[SUP]40)=3[SUP]40 hm...?
  5. O

    Solving for x using logarithms

    Homework Statement I tutor math for a couple of high school kids, and usually don't have any problems. Occasionally we run into a problem that takes me a minute, since I haven't actually used a lot of this stuff since I was in high school, but I always figure it out very quickly when that...
  6. D

    Expanding Logarithms with Radicals

    Homework Statement log2√xHomework Equations The Attempt at a Solution I thought that it might be something like log2x - log2x but that's not right. The book examples don't have any radicals.
  7. N

    Big-Oh algebra with logarithms that I don't get?

    My textbook says O(3log2 n) can be written as O(nlog2 3). Why is that? Thank you.
  8. N

    How Does the Principal Argument Affect the Solution of Log(exp(10i))?

    Homework Statement Log(exp(10i)) The Attempt at a Solution Since we want to get this in terms of the principal argument, would it matter if we ultimately wrote i(10 - 3\pi) or i(10 - 4\pi), since they are both in range?
  9. C

    Are logarithms a necessary evil in solving complex problems?

    I don't know why but when I'm solving a huge problem and I end up having to use logarithms I find it frustrating since they take most of my time up since I haven't studied them very often, and they're not very interesting in my opinion. What's your opinion on logarithms? Do they appeal to...
  10. S

    Help with logarithms and graphs

    1. Determine the equation y=f(x) for each of the following cases in simplest form. All of the plots are straight lines and all coordinates are indicated with abscissa first, ordinate second (x,y). -log y versus log x on a rectilinear graph passes through (5,7) and (2,3) -(y-2)^2 versus x...
  11. C

    Calculators How can I simplify this logarithmic equation on a TI89 titanium?

    First of all I would like to share this because I randomly found it and I commonly see that people approach base changes of common logs on the ti89 titanium in a very different way. Scroll half way down page 2...
  12. N

    Derivative of an exponential function using logarithms (lon-capa)

    Homework Statement What is the derivative of y=x^(13/x^2) with respect to x? The Attempt at a Solution I went through multiple techniques to solve this, but all of them have failed so far ._. In my latest attempt, I took the natural log of both sides: lny= lnx^(13/x^2) I...
  13. G

    Compound Interest Formula and Natural Logarithms

    Homework Statement Solve the compound interest formula for r by using natural logarithms. Homework Equations A=P(1+r/n)nt The Attempt at a Solution 1400 = 1000(1+r/360)(360*2) 1.4 = (1+r/360)720 ln(1.4) = 720ln((360+r)/360) I'm not sure where to go after this. Did I...
  14. T

    We're working with logarithms of base 3

    Homework Statement We're working with logarithms of base 3, and log(4)=a and log(7)=b. The goal is to put log(21) in terms of a and b. For example, take the log(112). It's the same thing as 2a+b since 4*4*7 = 112. Homework Equations Just the standard log properties. The Attempt...
  15. S

    MATLAB Matlab, logarithms and rounding small numbers to zero

    I'm a novice at MATLAB so I apologize if this is a dumb question: I need to find the sum of A, B and C given X, Y and Z, where ln(A)=X, ln(B)=Y etc. However, the values A, B and C are so small that when I try to use 'exp' the result is rounded to zero. This is a problem because later in my code...
  16. P

    Euler's Formula and Complex Logarithms relationship

    I've become rather curious, as of late, about the realm of complex logarthims; more specifially logarithms in the form log(z) where z is any negative number. Excuse any ignorance on my part, as I'm only in Precalculus, but I was just curious to see how Euler's formula is related to complex...
  17. P

    Can Logarithms Be Defined for Negative Numbers in Advanced Mathematics?

    I recently had a test (precalc) where we had to solve log(x)-log(x+4)=2 for x. The answer comes out negative. I understand that in precalc we are defining the logarithms for just positive numbers, but- Is it ever justified to define a logarithm for all numbers, both negative and...
  18. C

    Properties of Logarithms, Division and Multiplication

    Homework Statement Express in terms of logarithms x, y, z or w. Problem: loga(x8w/y2z4) Homework Equations log(u/w) = log u - log w log(uw) = log u + log w The Attempt at a Solution Here are my attempts: As you can see, the answers are pretty similar. I'm assuming...
  19. S

    Solving Simultaneous Equations Using Logarithms

    Homework Statement Solve the simultanious equations 2log_2y = log_43 + log_2x 3^y = 9^x The Attempt at a Solution 3^y = (3^2)^x y = 2x 2log_2 x + 2log_2 2 = log_2 \sqrt{3} + log_2 x log_2 x = log_2 \frac{\sqrt{3}}{4} No idea what to do from where, what does x equals?
  20. C

    Particle Distance/Velocity - Natural Logarithms

    Homework Statement I'm going to post an image due to the complex syntax in the problem... The Attempt at a Solution I'm not going to lie... I really have no idea where to even begin with this problem. Because it says total distance traveled by the particle, I'm assuming that the...
  21. B

    How Do You Express the Number of Security PINs in Logarithmic Form?

    Homework Statement A security PIN code is four digits long. Each digit could have the value 0 to 9. if an office building needs 10 security PINs, express the number of codes in logarithmic form with base 10 The Attempt at a Solution y=4log(x-10) I don't know how to do the...
  22. LiHJ

    Find the exact value of x Logarithms

    Homework Statement Find the exact value of x if: Homework Equations (3x)lg3=(4x)lg4. The Attempt at a Solution 3lg3xlg3=4lg4xlg4 (xlg3)/(xlg4)=(4lg4)/(3lg3) xlg3-lg4=(4lg4)(3-(lg3)) xlg(3/4)=(4lg4)(3lg(1/3)) Please help me, I am stuck here!
  23. S

    Finding the Solution for x in a Natural Logarithm Equation

    Homework Statement Solve for x xln(x)-24+6ln(x)-4x=0 My attempt at the solution I first moved "-24" and "-4x" to the right side of the equation yielding xln(x)+6ln(x)=24+4x I then converted the natural logarithms to exponent form and product form yielding ln(x)^x+ln(x)^6...
  24. D

    Solve Logarithm Math Problem: LOG3(X+3)+LOG3(X-1)=1

    Homework Statement NEED HELP WITH THIS MATH PROBLEM: LOG3(X+3) + LOG3(X-1) = 1Homework Equations The Attempt at a Solution I SAID LOG3(X+3) + LOG3 (X-1) SIMPLIFIES TO LOG3(X+3)*(X-1) = 1 I.E LOG3(X^2 +2X - 3) = 1 THEN 3^1 = (X^2 +2X - 3) AND 0 = (X^2 +2X - 6) THEN USE THE QUADRATIC FORMULA TO...
  25. N

    Help with two logarithms question

    Homework Statement NEED EXPLANATION HOW TO SOLVE QUESTIONS 46 AND 47 http://img685.imageshack.us/img685/2259/101020114327.jpg Uploaded with ImageShack.us Homework Equations The Attempt at a Solution
  26. M

    What's the point of using logarithms when sketching the inverse of a function?

    What's the point of logarithms when trying to sketch functions? Isn't y=3^x the same as x=3^y? I think it should be the same, but I get different results for each method. If it is the same, what's the point of y=log3x? It's confusing and I don't get the same results when trying to find the...
  27. K

    Ratio of logarithms in various bases to other bases

    I'm unsure as to if I am using the correct terminology, but what I mean by this is log = logarithm in base 10. ln = logarithm in base e. logx = logarithm in base x. Upon some investigation, I found that log(a)/ln(a)=log(b)/ln(b) where a and b are constants, meaning that there is a ratio...
  28. M

    A Question about Defining Logarithms as integrals

    When Logarithm are Defined as integral and the Exponetial functions are defined to be its inverses , then What can prove ? or why ? a^n = \underbrace{a.a.a...a}_{n-times} : n\in N also Why we define rational exponents as roots? I am sorry If my Question is silly. IS my Question...
  29. B

    Integral of Logarithms + Trig Functions

    Homework Statement Sec(x)/((ln(tan(x)+sec(x))^1/2) We were instructed to find the integral Homework Equations Here is a link to the wolfram solution, i don't understand the steps they...
  30. P

    Calculating Logarithms by hand in 1969

    This question is not about how to calculate the logarithm, but rather what method would be expected of someone in 1969. I am going through Apostol's Calculus, and in section 6.10 Apostol introduces polynomial approximations to the natural logarithm. Specifically, he introduces the following...
  31. M

    Finding Inverse Hyperbolic secant in terms of logarithms ?

    The Problem is when I Compute the Inverse I have to solutions sech^{-1}(x) = ln(\frac{1\pm \sqrt{1-x^{2}}}{x}) : 0<x\leq 1 And this not function which of them I will choose Another Question is how can I prove without the graph that csch (x) is one - to -one thanks
  32. J

    How Do You Simplify Logarithmic Expressions in Combinatorics?

    Homework Statement Show that ln[ (N + M - 1)! /M! (N-1)! ] is equal to N ln((N+M) / N) + M ln((N+M) /M). Homework Equations Using stirling's formula ln N! ~ N lnN - N The Attempt at a Solution ln[ (N + M - 1)! /M! (N-1)! ] (a) = (N+M...
  33. A

    Calculating Logarithms Before Calculators: History & Methods

    How were logarithms calculated before the use of calculators.
  34. L

    Can logarithms be applied to Modular arithmetic

    I was just curious. I believe the answer would be no, but I don;t know why
  35. S

    Exponents and Logarithms, equation

    Hey guys I need help how to solve this equation... Express your answer to the equation in the form alnb 9e^4x-e^2x=0 This is as far as I got 9e^4x=e^2x ln(9e^4x)=ln(e^2x) the answer given in the markscheme is x=1/2ln1/9, x=-1/2ln9, x=ln1/3, a=-1/2 and b=9, x=-ln3 (accept a=-1 and...
  36. Telemachus

    Is there an algebraic method for finding roots involving logarithms?

    Homework Statement Perhaps this is trivial, but working in thermodynamics oftenly requires of handling with logarithms. For example, in an exercise I'm trying to solve, I need the roots for something like: x+ln (x)=constant The thing is I don't know how to find this kind of roots. I mean, I...
  37. O

    Understanding Logarithms: Solving Equations and Creating Tables of Values

    Homework Statement Ok I'm taking an online advanced function course, and we are on logarithms and I don't get it one bit , there two questions given to me 1. y = 2 log10x and y = log5 (x – 2), Homework Equations The told me to make a table of value , i don't understand how i will get the...
  38. A

    Compound Interest Formula and Natural Logarithms

    Homework Statement Solve the compound interest formula for t by using natural logarithms. Homework Equations A=P(1+\frac{r}{n})^{nt} The Attempt at a Solution I start by dividing both sides by P. I then take the natural log of both sides and end up with ln(\frac{A}{P})=nt *...
  39. A

    Using Natural Logarithms to solve for x

    Homework Statement Use Natural Logarithms to solve for x in terms of y y = \frac{e^{10x}+e^{-10x}}{e^{10x}-e^{-10x}} Homework Equations I am not too sure. The Attempt at a Solution I multiplied both sides by the denominator first. Then I multiply by an LCD of e^{10x} I end up...
  40. L

    How you get results for logarithms

    Good morning everyone. Can someone please explain to me how you get results for logarithms. I understand that 10 power of 3 = 1000 but how do I work it out as Log10,1000. I know this is a strange request, but I can't get my head around it. Thank you
  41. Hepth

    Mathematica MATHEMATICA : Forcing Logarithms to Simplify

    How can I force : A Log[b] + A Log[c] to simplify to A Log[b c]? I tried, A,b,c all elements of reals, but it doesn't do it. (Assume everything is Real) I know I can make some patterned rules but I'd rather have it be by default. (i.e. : a_ Log[b_] + a_ Log[c_] :> a Log[b c])
  42. G

    Does Convergence of ln(a) to ln(b) Imply a Converges to b?

    Hi, If I have ln{a} converging to ln{b}, can I claim that a converges to b? I would think yes, because ln is a monotonic function and we are always sure to get a unique value of ln for any argument 'a'. Is that reasoning correct? I am not able to prove it by the definition of...
  43. T

    Hi, I have a problem. Its logarithms.

    325*(0.8)^t=5 Can anyone help me please?
  44. M

    Equations of Tangents to ln x at x = 1/2 | Logarithm Homework

    Homework Statement Find the equations of the tangents to the following graphs for the given values of x. (a) y = ln x, where x = 1/2 Homework Equations The Attempt at a Solution I know ln x differentiated is 1/x but I cannot see when the rest fall into the place. The book I'm...
  45. G

    What is the explanation for the complex logarithms function title?

    Homework Statement When was trying to solve this problem cos(x)=-2 I got the following answers i ln(2 +/- sqrt(3)) + 2 pi n n set of integers positive and negative now I was told by a couple of people on here to add 2 pi n to my solution because before it didn't have it. At the time I...
  46. Q

    Dimensions in logarithms after integration

    Homework Statement Let v = 1 / kt v = m/s k = 1/m t = s v = dx/dt so dx = dt / kt integrating, x = ln (kt)/k + C However the argument of a logarithm is dimensionless. But an integration is a perfectly normal thing to do. So how come this integration results in a...
  47. X

    Logarithms and Index Laws help

    !Logarithms and Index Laws help! Ive been trying to complete a revision sheet but it has logarithims and index laws, and i don't know how to do them. I've included a few of the hardest questions frome the revision sheet in the attachment. ...Look at attachment.... I've tried the...
  48. J

    Simplifying an Expression using Logarithms

    Ok, I have an Expression which is attached and it needs simplifying. What I have done is got 12logx But I don't think this is correct. Another possible place I got to is logx4/logx1/3 Which one is it?
  49. J

    Simplifying Expression using Logarithms

    Simplify the attached expression using any relevant logarithmic rules I haven't really done much with logarithms, so i didnt know where to start?
  50. E

    Calculus, Integrals with Natural Logarithms

    Homework Statement ∫tan^2(2x)/sec2x dx; u=sec2x; du=1/2tan^2(2x)dx. Homework Equations ∫1/x(dx)-ln|x|+C. ∫1/u(du)=ln|u|+C The Attempt at a Solution This is me trying to rewrite the equation. (sin^2(2x)/cos^2(2x))/(1/cos2x), (sin^2(2x))/(cos(2x)). Honestly, I feel lost trying to...
Back
Top