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. caters

    How can I convert directly between two non-power-of-ten bases?

    I can most of the time successfully convert between base 10 and another base or another base and base 10 or between 2 bases where one of them is a power of the other(like base 2 and base 4 or base 3 and base 9). With negative bases I sometimes don't get what I want in that negative base and...
  2. R

    How Can Logarithmic Differentiation Prove dy/dx Equals y/x?

    the variables x and y are positive and related by x^a.y^b=(x+y)^(a+b) where a and b are positive constants. By taking logarithms of both sides, show that dy/dx=y/x. provided that bx not equal to ay.
  3. P

    Point of Intersection involving logarithms

    Homework Statement Solve for the point of intersection between ##y=\log_{2}{(2x)}## and ##y=\log_{4}{(x)}## Homework Equations 3. The Attempt at a Solution [/B] Setting the two equations equal: $$\log_{2}{(2x)}=\log_{4}{(x)}\\ 2x=2^{\log_{4}{(x)}} \\ 2^{2x}=4^{\log_{4}{(x)}}\\...
  4. Christo

    How Is Doubling Time Calculated in Population Growth Models?

    1. The population of a certain country grows according to the formula: N = N0e^kt Where N is the number of people (in millions) after t years, N0 is the initial number of people (in millions) and k = 1/20 ln 5/4. Calculate the doubling time of this population. Leave your answer in terms of ln...
  5. Chrono G. Xay

    Math Trouble in Physics Land: Logarithms

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  6. C

    Strange Pattern with Logarithms

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  7. anemone

    MHB Minimum of the Sum of Logarithms

    Find the minimum of $\large \log_{a_1}\left(a_2-\dfrac{1}{4}\right)+\log_{a_2}\left(a_3-\dfrac{1}{4}\right)+\cdots+\log_{a_n}\left(a_1-\dfrac{1}{4}\right)$ where $a_1,\,a_2,\cdots,a_n$ are real numbers in the interval $\left(\dfrac{1}{4},\,1\right)$.
  8. J

    Can Euler-Lagrange Equations Explain Mirages?

    Homework Statement On very hot days there sometimes can be a mirage seen hovering as you drive. Very close to the ground there is a temperature gradient which makes the refraction index rises with the height. Can we explain the mirage with it? Which unit do you need to extremalise? Writer the...
  9. M

    Complex Analysis: Largest set where f(z) is analytic

    Homework Statement Find the largest set D on which f(z) is analytic and find its derivative. (If a branch is not specified, use the principal branch.) f(z) = Log(iz+1) / (z^2+2z+5) Homework EquationsThe Attempt at a Solution Not sure how to even attempt this solutions but I wrote down that...
  10. 22990atinesh

    Small confusion regarding logarithmic formula

    I've a small confusion about formula ##\log^3 n = \log \log \log n## or ##\log^3 n = (\log n)^3 ##
  11. B

    Are Logarithms Useless? Why Need them Given Exponents?

    Just curious, since we're discussing them this week in my class. Why do we need logarithms? Aren't they just a concoction to express exponents?
  12. G

    Solving a Math Problem: Demystifying the Unknown

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  13. alyafey22

    MHB Higher order reflected logarithms

    Let us define the following $$I(n,m) = \int^1_0 \log^n(x)\log^m(1-x)\,dx$$ Our purpose is finding a closed form for the general case. Note: for a given n and m the above formula can be deduced by succesive differentiation of the beta representation $$B(p,q) = \int^1_0 x^{p-1}...
  14. I

    Logarithms and scientific calculator

    Homework Statement y= -2log3(x-3) -1 Homework Equations I am using a SHARP EL-546W scientific calculator, and I do not know what steps to take in order to find a point given an x value. i.e. if x=3, then y=6. I cannot seem to get 6 on my own and I have tried a wide variety of methods and...
  15. Rectifier

    How Does the Logarithmic Equation Simplify with Large Frequency Values?

    Hey! I have a question regarding a statement in my physics book. I don't see how |H(f)|_{dB} = - 10log (1+( \frac{f}{f_B})^2) approaches this equation below for big values on f. |H(f)|_{dB} = - 20log ( \frac{f}{f_B}) Could you please help me out? Thanks in advance. EDIT: I am sorry if...
  16. C

    Ph Scale Logarithms: Calculating Acidity Difference

    How do I work out how many times more acidic a ph value of 6.8 is compared to 8.6? For example.
  17. Maxo

    Different ways of expressing logarithms

    Homework Statement Show that ^{2}log(e)=\frac{1}{ln2} Homework Equations ^{a}log(x) = ^{a}log(b)\cdot ^{b}log(x)^{a}log(x)=\frac{^{b}log(x)}{^{b}log(a)} The Attempt at a Solution How can this be shown? I assume it can be done just using logarithm laws, but I don't see how. I tried manipulating...
  18. C

    MHB How Can I Express a Combination of Logarithms as a Single Logarithm?

    Express as a Single Logarithm. Assume all the logarithms have the same base. The problem is: $4*\log_{d}\left({a}\right)-\frac{5}{6}*\log_{d}\left({b}\right)+\frac{2}{3}*\log_{d}\left({c}\right)$, where $d\gt1$ then I use the power rule for logarithm for the beginning and use some grouping...
  19. wolf1728

    Terminology for exponentiation and logarithms

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  20. L

    Will logarithms help in calculating this product?

    Homework Statement Calculate the value of y in the expression below: 10y = 103.2 × 102.4 × 10-1.8 × 1000.3 × 100-0.5 Homework Equations The Attempt at a Solution 10y= 103.8*100-0.2 Don't know how to move on from here as have no other examples like it? I am thinking I can use logs, however...
  21. N

    Difference between Logarithms and Exponentials

    Hello, I am beginning to learn precalculus. I understand that there are times where you can change logarithms to exponential expressions. So how are they different and similar and why are they interchangeable?
  22. M

    Horribly Confused With Complex Logarithms

    I'm getting myself all confused with complex logarithms. I'll try to explain why. One identity with complex logarithms is ln(z^c)=cln(z)+2πik, with k an integer. This is, of course, a more general case of ln(e^c)=c+2πik, but it doesn't always work the same! Let's say we are evaluating ln(e^i)...
  23. Rectifier

    Increasing or decreasing - logarithms

    Hey there! How do I determine whether this function is growing or decreasing without using any calculator or graphing tools? y=ln(\frac{1}{x}) I know that y=ln(\frac{1}{x}) \ \Leftrightarrow \ e^y=\frac{1}{x} \ \ \ (1) Then I tried to use the formula f(x)=a^x and then determine whether...
  24. S

    MHB How can the properties of logarithms be used to simplify and solve equations?

    Use the properties of Logarithms to write the expression as a sum, difference, and/or constant multiple of logarithms:
  25. D

    How to understand Logarithms, Fundamentally

    Alright so my problem is if I don't see exactly how things work (in math) I can't use them. Like what I mean is some people use memorization and other methods but I really need to understand things deep down to be successful. So my problem is, I have trouble understanding logarithms. Now I...
  26. N

    How can Logarithms be applied to real life situations and examples?

    If I understand Logarithms correctly it is the orders of magnitude? Either exponentially growing or exponentially decaying. I've heard that exponential growth is the same as the growth of cancer cells. But what are some other real life applications/examples of logarithms? Thank You
  27. S

    Geometric series involving logarithms

    Homework Statement A geometric series has first term and common ratio both equal to ##a##, where ##a>1## Given that the sum of the first 12 terms is 28 times the sum of the first 6 terms, find the exact value of a. Hence, evaluate log_{3}(\frac{3}{2} a^{2}+ a^{4}+...+ a^{58}) Giving...
  28. N

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    Hello, I have been studying Logarithms in University. I understand it's how many of ONE number to get another number, and I see how it is rearranged to find these "missing" links. But maybe I am overlooking something, but I don't quite see the bigger picture here with how to use Logarithms. How...
  29. L

    Proving y= 10^x Using Logarithms to Base e

    Homework Statement If y= 10^x, show by taking logarithms to base e that y = exln10 Homework Equations The Attempt at a Solution Well what I did was y= 10^x so ln y = xln10 they told me that y = exln10 so ln y = xln 10 , so ln y = lny so y = y so y= 10^x. The way I...
  30. C

    Solving Logarithms: Find x in 2x^2-7x+3 = 0

    Homework Statement Log(2x-3)=log(4x-3)-logx Homework Equations Logcl =logcr L=r The Attempt at a Solution Quotient law of logs So I simplfied: Log(2x-3)=log(4x-3)/x Since same base: 2x-3=4x-3/x 2x^2-7x+3=0 (X-1)(x-6) The ans is supposed to be 3. I don't know what I'm doing wrong?
  31. A

    Solve 2^3x=7^(x+1): Logarithms Solution

    Homework Statement 2^{3x} = 7^{x+1} Homework Equations N/A The Attempt at a Solution 2^{3x} = 7^{x+1} I'm not entirely sure how to proceed from here. I'm assuming that the following is the right step towards the solution... (log2)(3x) = log(7)(x+1) ...but I'm not...
  32. H

    Negative Numbers and Logarithms: Is it really wrong?

    I know that in logarithms we can not set as base a negative number,but look at this(in the brackets I will put the base.): log(-2)-8=3 Mathematics say that is wrong,but why? If we tell -2^3=-8 we have a correct result. So? Thank you!
  33. A

    When to add and cancel logarithms

    I don't have a single homework question but a general one that will help me out tonight. I am taking calculus one college level and we are reviewing logarithms and I am really having trouble. I just don't understand how we are supposed to know where to add and cancel them. It seems that at...
  34. H

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  35. M

    Are logarithms only non-negative?

    I read that logb x exists only when x >= 0 what about log-3 -27 though? The answer should be the exponent 3, right? Thanks!
  36. M

    Can you distribute logarithms?

    Does anyone know if the following is true: logb (x + y) = logb x + logb y Thanks. This isn't homework, but I am just wondering if the following is true. I already know the logarithm product, quotient, and power rules!
  37. interhacker

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    Homework Statement If a solution containing a heavy concentration of hydrogen ions(i.e., a strong acid) is diluted with an equal volume of water, by approximately how much is its pH changed? (Express (pH)diluted in terms of (pH)original.) Homework Equations I think the question...
  38. T

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  39. A

    Where can I find a good logarithms book or PDF for summer study?

    so i want to study logarithms this summer , i'd like you guys to recomonned for me a good book or site i used to go to this one http://tutorial.math.lamar.edu/download.aspx it really helped me understand calculus I but they really lack info on logs , thanks in advance.
  40. D

    Simple question about natural logarithms

    ln(a * b) = ln(a) + ln(b) and ln(a/b) = ln(a) - ln(b) correct?
  41. T

    Rules of Logarithms: Explained and Demystified

    Hi, So I'm doing boltzmann's entropy hypothesis. I have a basic question about the mathematics of logarithms. For \frac{ΔS}{K_B}=ln(W_f)-ln(W_i), I do the correct maths and go \frac{ΔS}{K_B}=ln(\frac{W_f}{W_i}), and finally take the log of the equation to get...
  42. J

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    The math behind the logarithms isn't hard especially if you can remember where things go and how the math is worked out but what are they for? What do they do and why are they used? Maybe I am missing what they are used for in the math but it really just doesn't make sense. You have a base...
  43. R

    Quick question about exponential and logarithms

    If I have ln(e^(-8.336/10c)) wouldn't that be the same as ln(e^(1/(8.336/10c))) therefore = 1/(8.336/10c) = 10c/8.336? I am confused about this because in my lecture notes they simplified ln(e^(-8.336/10c)) to just = -8.336/10c :confused: Your help would be appreciated!
  44. M

    Finding derivatives of inverse trig functions using logarithms

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  45. J

    Exponential Variable and Logarithms

    Homework Statement I'm trying to find the value of a variable which happens to be an exponent. Homework Equations 230,000=1500\frac{(1+.00077)^n-1}{.00077} I believe I need to use logarithms to get to my answer, but I've reviewed logarithm rules and I'm stuck. The Attempt at a...
  46. J

    Solving Logarithmic Equations: Analytical Method

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  47. Mandelbroth

    Logarithms in Systems of Equations

    Homework Statement Consider the system... y=e^{-x}+1 \\ y=2+lnx The Attempt at a Solution I don't think there is a "pretty way" (algebraic manipulation) to do this. I simply found the point where the plots intercepted, getting the solution set (0.62745018..., 1.5339409...). Is there a...
  48. Astrum

    Reviewing my logarithms, solving this inequality

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  49. JJBladester

    Logarithms - Show that the expression is true

    Homework Statement Show that the following expression is true: 0.2*0.8 = -14dB + (-1.94dB) Homework Equations 1 dB = 10log10(x) (where x is a ratio of two quantities) 10log10(ab) = 10log10(a) + 10log10(b) The Attempt at a Solution 0.2*0.8 = -14dB + (-1.94dB) 0.16...
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