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.

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  1. T

    Help with Logarithms: Expressions in Terms of r, s, & t

    Homework Statement Let r = log9 31, s = log6 31, t = log9 6. Write the following expressions in terms of r, s, and/or t. The change of base formula may be helpful in finding some of these logarithms. Homework Equations log 9 base 31 The Attempt at a Solution I am not sure where...
  2. C

    What Are the Solutions to These Calculus Problems?

    hints? Derivatives: Intervals, stationary points, logarithms, continuous functions Homework Statement Got any hints or anything? 1. Suppose that f(x) = (x - 3)^4 ( 2x + 5)^5 a) Find and simplify f ' ( x ) b) Find stationary points of f c) Find exactly the intervals where f is...
  3. E

    Proofing Logarithms: a^(log(b))=b^(log(a))

    Does anyone know how to proof the following: a^(log(b))=b^(log(a)) for a,b>0
  4. D

    Solve Logarithms: Find X When 8logx-3logx^2=log2

    logarithms STUCK! 8logx-3logx^2 = log8x - log4x im struggling to find x. my working out so far is: rhs log8x-log4x = log2 = 8logx - 3logx^2 = log 2 = 8logx - 3logx^2 - log 2 = 0 = -b+ or - the square root of b^2-4ac divided by 2a a = -3 b = 8 c = - 2 = -8 + or - sqaure...
  5. M

    Complex Logarithm: Solving tan-1[(2sqrt(3) - 3i)/7]

    Homework Statement If tan-1z = (1/2i)ln[(1+iz)/(1-iz)] then find tan-1[(2sqrt(3) - 3i)/7] The Attempt at a Solution I haven't gotten very far, but this is what I have so far: tan-1[(2sqrt(3) - 3i)/7] = (1/2i)ln[(i2sqrt(3) + 10)/(i2sqrt(3) + 4)] Where do you go...
  6. James889

    Help with Logarithms: Find x When 2lnx=xln2

    I have 2lnx = xln2 where x\ne2 if you start by dividing both sides by ln2 is the following legal? \frac{2lnx}{ln2} \rightarrow x = 2ln(x-2) e^{2ln(x-2)} = (x-2)^2 x = (x-2)^2 \implies x = 4
  7. G

    Calculating Logarithms: \log_a (b) = {\ln (b) \over \ln (a)}

    I read that \log _2 (3) = {\ln (3) \over \ln (2)} Is \log _a (b) = {\ln (b) \over \ln (a)} How?
  8. B

    Indefinite Integration with Logarithms and Substitution

    Hi, I missed a few days of my calculus class. I've managed to figure out how to use substitution to solve an indefinite integral, and can apply the log properties to some extent. I just can't figure out this problem. Homework Statement Find the indefinite integral: \int{\frac{1}{x...
  9. X

    Integration by parts involving exponentials and logarithms

    Homework Statement Using integration by parts, integrate: (1/x^2)(lnx) dx with the limits e and 1 Homework Equations [uv]to the limits a b - the integral of (v)(du/dx) dx (sorry, don't know how to write out equations properly on a computer) The Attempt at a Solution I've...
  10. M

    Evaluating Logarithms: Solving 2^x - 2^{1-x} = 1 in Simple Steps

    Hello I was wondering if someone could help me. I've got a question on evaluating logarithms, but I've not done anything like this before. I'm only used to the basic logarithmic stuff, and I still find that a bit confusing. So, I've been trying to do this question for hours now, and...
  11. R

    Logarithms and online integrators

    The integral of ln(1-x) is -(1-x)ln(1-x)-x, when 0<=x<=1. So for example: \int_{0}^{1} ln (1-x)dx= (-(1-x)ln(1-x)+x)_{0}^{1}=-1 However, going to an online integrating site: http://integrals.wolfram.com/index.jsp?expr=Log[1-x]&random=false they give the integral of ln(1-x) as...
  12. T

    How exactly do logarithms work?

    I was just wondering what's the "magic" behind logarithms. Just took them in school, and I get the part that they're just the reverse of exponents but it still feels sort of like a mystery every time I solve a logarithm using a calculator. Just like how addition is the slow way of...
  13. P

    Solve Natural Logarithm Equation lnx+ ln(x-1) = 1

    Homework Statement lnx+ ln(x-1) = 1 solve each equation for x Homework Equations ln(e^x) = x e^lnx = x The Attempt at a Solution x + (x-1) = e^1 [==> using ln(e^x) = x] from this point on, I am stuck because I am having trouble isolating x because of the x that is in the brackets...
  14. T

    Formula for finding logarithms possible?

    Is it possible to make a formula for logarithms of any base? logb(a)=x I want to find x through some formula. I've seen that you can use a series for e as the base, is that the only base that can be solved for? Is there any work being done to accomplish this, or, maybe it has been...
  15. I

    Reducing logarithms of factorials

    Homework Statement Part of a much bigger problem, but I am hung up on solving the following: ln\left [ \left(\frac{N+n}{2}\right ) ! \right ] = \left ( \frac{N+n+1}{2}\right) \frac{ln(N+n)}{2}\right ) I am trying to follow a proof in...
  16. J

    Logarithms questions need checking

    1. ln (x2 / y2) = log (A / B) = logA-logB = logx2-logy2 = 2logx - 2logy (is this correct) 2. can this expression be split into separate log functions and numbers, is this question the answer log(2x + 3y) = log(2x + 3y) (is this correct)
  17. J

    Exponential relationships to logarithms and straight line graph?

    it is suspected that cells in a sample are dividing so that the number of cells present at anyone time t (measured in seconds) is growing exponentially according to the relationship y = 64 x 2^2t. it would be hard to check this relationship accurately by plotting measurements of y against t, so...
  18. J

    Problem with Logarithms solutions

    I know why I can add 2 \pi i to log_{e} \omega ( log_{e} \omega = log_{e}z + i \theta if \omega = z(cos \theta +i sen \theta ) ) but I can't understand why I can add \frac{2 \pi i}{log_{e}b} to log_{b} \omega . Does anyone have the answer for me? Thanks!
  19. T

    Verifying Logarithms Homework: Get Help Here

    Homework Statement log(1/2xy^2) = ln x + 3 ln y - ln 2 Homework Equations The Attempt at a Solution i have been trying for hours on with next to no results even just an insight on how to go about it would be helpful
  20. B

    Solving a certain equation involving logarithms

    I'm reading book called "Prime Obession" which attempts to give a layperson's introduction to the Riemann Hypothesis. In laying the groundwork in one of the early chapters the author is explaining the fact that the function log(x) increases more slowly in total than x raised to any power. For...
  21. C

    Logarithms where you are solving for X

    Homework Statement Put in Exponential form and solve for X. SHOW YOUR STEPS! log2(x-5)=5 Homework Equations I am not sure what this is asking for. There are no other equations. The Attempt at a Solution 25=X-5 25+5=X-5+5 32+5=X X=37
  22. L

    Infinite alternating series with repeated logarithms

    Homework Statement Calculate the sum of the following series: \sum_{i=2}^{\infty}(-1)^i \cdot \lg ^{(i)} n Where (i) as a super-script signifies number of times lg was operated i.e. \lg ^{(3)} n = (\lg (\lg (\lg n))) , and n is a natural number. Homework Equations The Attempt...
  23. C

    Solve Natural Logarithms ln(x+1)+ln(x+3) < ln(x+7)

    ln(x+1) + ln(x+3) < ln(x+7) Find x! I should of course know this, second year at high school (in Norway). I've just forgotten the way attack the whole thing^^
  24. A

    Calculating logarithms by hand

    Is there a way to quickly calculate a log by hand? It needs to be "quick & dirty", much like other arithmetic done by hand (i.e. it doesn't have to be pretty, but needs to be quick and accurate, or at the very least, a very accurate estimate). The only thing I can think of is trial & error...
  25. D

    Solving Exponential Equations with Logarithms: Restrictions and Techniques

    Homework Statement Solve for x. State restrictions, if necessary. 3^(2x) - 3^(x) -12 = 0 Homework Equations The Attempt at a Solution 2xlog3 - log3 = log 12 x=log 12/log 3 Doesn't work. I have no idea how to do this.. we didn't learn it. The 2x is throwing me off.
  26. M

    How Do You Solve Exponential Equations Using Logarithms?

    Homework Statement 12^x=4X8^(2x) Big X= multiplication sign little x= unknown i simply cannot figure this out. Any help please? Homework Equations 4.6X1.06^(2x+3)=5X3^(x)
  27. R

    Differentiation of natural logarithms

    I encountered a proof problem when I was reading up on the derivatives of natural logarithms' section. It gave a rule which said this : \text{For } a >0 \text{ and } a\ne 1 \text{,}\\\frac{d}{dx}(a^{u}) \ = \ a^{u} \ \ln{a\frac{du}{dx}} To prove it on my own, I made a few identities: a^{u}=y...
  28. R

    Solving Limit Involving Logarithms

    Hi! I'm having trouble solving this limit: lim x->infinite ln(1+2^x)ln(1+3/x) Revelation
  29. N

    Mastering Logarithms: Solving Equations with Logarithmic Functions

    Homework Statement Im not looking for answers as i know the answers already. i also know that given time and a calculator, i could figure these out, but i have neither for the exams. I am just wondering the general way of setting problems like this up, so i can solve others in the future...
  30. P

    Induction question with logarithms

    I have to prove that 2^n > n^2 for every n>=5 So... 2^k > k^2 /log base 2 log2(2^k) > log2(k^2) k*log2(2) > 2*log2(k) k/2 > log2(k) So I'm stuck here and I am having problems solving for k, since i have it on both sides. I just need someone to gimme a slight push :)
  31. P

    What are some strategies for solving logarithmic equations?

    I have two problems if that's ok with you 1. log((x-3)/(x+3)) <= 1 2. 6*9^(1/x) - 13*6^(1/x) + 6*4^(1/x) = 0 So for the first one i didnt get too far. I divided the logarithm and i got: log(x-3) + log(x+3) <= 1 For the second one i used logarithms and received: log6 + 1/x*log9...
  32. E

    How to Solve Logarithmic Equations with Different Bases?

    Solve the equation log2x + log4x= 5. To start, should I change this to an exponential... I am stuck because I have only done log questions that have the same base.
  33. T

    Prove the existence of logarithms

    Fix b >1, \ y >0 , and prove that there is a unique real x such that b^{x} = y . Here is the outline: (a) For any positive integer n , b^{n}-1 \geq n(b-1) . Why do we do this? (b) So b-1 > n(b^{1/n}-1) . (c) If t>1 and n > (b-1)/(t-1) then b^{1/n} < t . etc.. Is...
  34. S

    Logarithms: ALG2 teacher say what?

    I have these questions that are due tomorrow I am completely clueless on what my teacher is asking. 1. What is log(x), Explain. I think that is like a parent function not sure 2.What is 10^log(10) I know that it graphs as a straight line but that's it...
  35. Y

    What Is the Intensity of Sound at 123 dB?

    Logarithms and Intensity---HELP ASAP! Homework Statement a)What is the intensity of sound at 123 dB? b)Compare it to that of a whisper at 20 dB. Homework Equations something with logarithms?? The Attempt at a Solution a) 10^123
  36. P

    How can logarithms help solve an equation involving trigonometric functions?

    4^{sinx} = \sqrt[cosx]{2} I use logarithmic and i get : sinx * log(4) = (1/cosx) * log(2) sinx*cosx = log(2)/log(4) what should i do next?
  37. P

    Are Common Logarithms Explained in Your Textbook?

    what are common logarithms? my textbook doesn't explain it very well
  38. M

    Real & Imaginary Parts of Logarithms

    Homework Statement Find the real & imaginary parts of log(1+i)log(i) ? Homework Equations The Attempt at a Solution
  39. T

    Do Logarithms play any major part in being able to do calculus?

    do Logarithms play any major part in being able to do calculus?
  40. K

    How Do You Differentiate Products of Logarithms?

    [SOLVED] derivative with logarithms Ok, so the problem problem probably isn't as bad as I'm making it, either that or its because its getting late & my brain just isn't functioning. Find the derivative of y with respect to r. y=log _2 \left( r \right) * log _4 \left( r \right) The...
  41. K

    How Do You Integrate Using Logarithms for \(\int \frac{3}{3x-2} \, dx\)?

    Evaluate the integral. \int3/(3x-2) dx from 0 to -1 (top to bottom). I change the equation to [tex(1/x - 3/2) dx[/tex] then integrated ln x-3/2x, but ln x at 0 is undefined. The textbook shows it as becoming ln (3x-2), but I'm not completely understanding how to get to that.
  42. M

    Solving Logarithms Questions - Matt's Attempt

    Just a quick check, I've spend hours trying to crack this. Have I gone the right way about it? Homework Statement Solve 5^(x-1)= 4^(1-3x) Homework Equations The Attempt at a Solution (x-1)log5 = log4 (1-3x) xlog5 - log5 = log4 - 3xlog4 xlog5 + 3xlog4= log4 + log 5...
  43. A

    Logarithms and Exponents Question

    [SOLVED] Logarithms and Exponents Question Homework Statement 5^{x}=41 The attempt at a solution Well, I know that one way to figure this out would be that to find a common base for both sides of the equation and then use the known exponent to find the variable. The only thing...
  44. P

    Differentiating logarithms and expoentials

    1. Differenciate the following functions i) y = x2ln(4x) ii) y = ln (x + 1)/x iii) y = ln (x2 - 1)1/22. Laws for differentiating logs and exponentials 3. I did some of the more easy one's, these ones just got me stumped. i) i think you would use the product rule. so u = x2 and v =...
  45. X

    Logarithms disinfectant spray problem

    Question: A new disinfectant spray is expected to kill 50% of the known germs in a room, but for health reasons it can only be used once a day. Between spraying, the germs increase by 25%. How many consecutive days of spraying are required to reduce the germs in the room to 10% of the original...
  46. B

    Can the Power Rule be applied to all rational numbers in logarithms?

    All information, including the problem, is attached. So far I think I've proven by induction that log (a^r) = r log (a) whenever r is an integer, but I need to prove this for all rational numbers r = p/q . We're working with the functional equation that has the property that f(xy) = f(x)...
  47. P

    Solving Natural Logarithm Equation ln(x^2 + 1 ) -3lnx=ln2

    Homework Statement ln(x^2 + 1 ) -3lnx=ln2 Solve for x. Homework Equations The Attempt at a Solution I used laws of logarithms to simplify it down to -x^2(x-1)=1 . I don't think this is the answer I'm under the impression you need x = for the answer. First I brought the 3...
  48. A

    How Do You Solve Logarithms with Square Roots?

    I've been working on logarithms in my calculus class and I came acrossed one with a square root. Since I am new to these types of problems I'm not sure how to work with this yet. How would I solve something such as loge[square root of e]
  49. O

    How to Solve Logarithmic Equations Quickly and Easily?

    Homework Statement 3^(x - 2) = 30 (x - 2)^3 = 30 log (base 4) (x + 3) + log (base 4) (x - 3) = 2 the square root of: (4^(x + 3) / 16^x) = 32 log (base 9) x = 1.5 Homework Equations NONE The Attempt at a Solution I haven't done this since last year. Someone refresh my...
  50. M

    Solve Simple Logarithms: log91/27

    Homework Statement #8: Simplify: log91/27 Homework Equations y=logb(x) x=b^y The Attempt at a Solution I have personally not tried logarithms before, but I think the problem is really easy and I am probably just over-thinking it? I think that, using the formula's below, 9^a=1/27 So...
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