What is Logarithm: Definition and 337 Discussions

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:




{\displaystyle \log _{b}(x)=y\ }
exactly if




{\displaystyle \ b^{y}=x\ }


{\displaystyle \ x>0}


{\displaystyle \ b>0}



{\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:










{\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. tellmesomething

    Approximation used in physics explanation

    I dont know if this question is more fit for the physics forums but regardless i have a doubt in a suggested approach to these questions. So this is really easy right, you can do this simply by taking out the change in KE which comes out to be 0.0201 K (K being the initial KE) and then find...
  2. RChristenk

    Show that if ##x>1##, ##\log_e\sqrt{x^2-1}=\log_ex-\dfrac{1}{2x^2}-##

    ##\log_e\sqrt{x^2-1}=\dfrac{1}{2}[\log_e[(x+1)(x-1)]]=\dfrac{1}{2}[\log_e(x+1)+\log_e(x-1)]##. ##\Rightarrow \log_e(x-1)=\log_e[x(1-\dfrac{1}{x})]=\log_ex+\log_e(1-\dfrac{1}{x})## We know: ##\log_e(1+x)=x-\dfrac{x^2}{2}+\dfrac{x^3}{3}-\cdots##...
  3. RChristenk

    Logarithmic Series question for finding ##\log_e2##

    By definition: ##\log_e(1+x)=x-\dfrac{x^2}{2}+\dfrac{x^3}{3}- \cdots ## ##(1)## Replacing ##x## by ##−x##, we have: ##\log_e(1-x)=-x-\dfrac{x^2}{2}-\dfrac{x^3}{3}- \cdots## By subtraction, ##\log_e(\dfrac{1+x}{1-x})=2(x+\dfrac{x^3}{3}+\dfrac{x^5}{5}+ \cdots)## Put ##...
  4. brotherbobby

    Finding the domain (and range) of a logarithmic function

    Problem statement : I copy and paste the problem from the text. You will note that I added the range myself, because it seemed relevant and yet I couldn't do it. Attempt : I could evaluate the domain. The base of the function ##x-4>0\Rightarrow x>4.\hspace{60 pt} (1)## The function...
  5. chwala

    Solve the given trigonometry equation

    In my approach i have the following lines ##\ln (x + \sqrt{x^2+1}) = 2\ln (2+\sqrt 3)## ##\ln (x + \sqrt{x^2+1} = \ln (2+\sqrt 3)^2## ##⇒x+ \sqrt{x^2+1} =(2+\sqrt 3)^2## ##\sqrt{x^2+1}=-x +7+4\sqrt{3}## ##x^2+1 = x^2-14x-8\sqrt 3 x + 56\sqrt 3 +97## ##1 = -14x-8\sqrt 3 x + 56\sqrt 3 +97##...
  6. RChristenk

    Distributing negative signs in logarithms

    Simplify ##\log(A \times B \div C \times D)## Is it ##\log(A)+\log(B)-(\log(C)+\log(D))## or ##\log(A)+\log(B)-\log(C)+\log(D)##? I'm leaning toward the former but not sure. Thanks.
  7. RChristenk

    Find the value of ##\sqrt[5]{0.00000165}##

    ##\log x=\log\sqrt[5]{0.00000165}## ##\Rightarrow \log x =\dfrac{1}{5}\log0.00000165=\dfrac{1}{5}(\overline{6}.2174839## ##\Rightarrow \dfrac{1}{5}(\overline{10}+4.2174839) = \overline{2}.8434968## This is the solution I'm given. I don't understand the last line. First, why is...
  8. RChristenk

    Efficient Logarithmic Calculations for 0.3048 without a Calculator

    ##0.3048=\dfrac{3048}{10000}=\dfrac{2^3\cdot3\cdot127}{10^4}## ##\log0.3048=\log(\dfrac{2^3\cdot3\cdot127}{10^4})## ##\Rightarrow 3\log2+\log3+\log127-4\log10## I don't have the value for ##\log127##, and this problem is to be solved without a calculator. All the logarithms are base ##10##...
  9. Z-10-46

    A Even with a whimsical mathematical usage, solutions are obtained!

    Hello everyone, Here, we observe that the familiar properties of the real logarithm hold true for the complex logarithm in these examples. So why does a whimsical mathematical use of real logarithm properties yield coherent solutions even in the case of complex logarithm?
  10. A

    A Extending reals with logarithm of zero

    What do you guys have to say about this Mathoverflow post? Do you have any interesting ideas about this? https://mathoverflow.net/questions/432396/extending-reals-with-logarithm-of-zero-properties-and-reference-request
  11. brotherbobby

    Solving a nested logarithmic equation

    Problem statement : Let me copy and paste the problem on the right as it appears in the text. Solution : Using the Relevant Equations (2) and (3) above, we can claim that \begin{align*} &\log_{2x^2+3x+5}(x^2+3)=1\\ &\Rightarrow x^2+3 = 2x^2+3x+5\\ &\Rightarrow x^2+3x+2=0\\ &\Rightarrow...
  12. C

    Writing a logarithm in a form not involving logarithms

    logyx + logxy = 3/2 Solution $$\begin{align*}\log_{ y }{ x } + \log_{ x }{ y } &= \frac{ 3 }{ 2 } \\ \log_{ x }{ y } &= \frac{ \log_{ y }{ y } }{ \log_{ y }{ x } } \\ \log_{ y }{ x } + \frac{ 1 }{ \log_{ y }{ x } } &= \frac{ 3 }{ 2 } \\ \left(\log_{ y }{ x } \right)^ { 2 } + 1 &=...
  13. YouAreAwesome

    B Can a log have multiple bases?

    Hi, I tutor maths to High School students. I had a question today that I was unsure of. Can the natural log be to the base 2? The student brought the question to me from their maths exam where the question was: Differentiate ln(base2) x^2 If the natural log is the inverse of e then how does...
  14. G

    B Black Hole Entropy: Basis of Logarithm Explored

    In textbooks, Bekenstein-Hawking entropy of a black hole is given as the area of the horizon divided by 4 times the Planck length squared. But the corresponding basis of the logarithm and exponantial is not written out explicitly. Rather, one oftenly can see drawings where such elementary area...
  15. chwala

    Can Logarithms Be Used to Solve a Tricky Exponential Problem?

    Find question here; See my attempt; $$a^x=bc$$ $$(a^x)^{yz} = (bc)^{yz}$$ $$a^{xyz} = (b^y)^z (c^z)^y $$ $$a^{xyz} = (ac)^z (ab)^y $$ $$a^{xyz} = a^z c^z a^y b^y$$ $$a^{xyz} = a^z(ab)a^y(ac)$$ $$a^{xyz} = a^2(bc) a^{y+z}$$ $$a^{xyz} = a^{y+z+2} (bc)$$ $$a^{xyz} = a^{y+z+2} a^x$$ $$a^{xyz} =...
  16. chwala

    Find the value of ##N## in the logarithm problem

    My approach is as follows; $$\log_8 N= \frac {1}{2} p$$ $$\log_2 (2N)=q$$ $$→8^{\scriptstyle\frac 1 2} = N$$ $$ 2^q=2N$$ $$2^{\scriptstyle\frac 3 2} =N$$ $$2^q= 2N$$ then from 1 and 2, it follows that, $$2^{q-1.5p} =2,$$ on solving the simultaneous equation; $$q-1.5p=1, q-p=4$$, we get...
  17. chwala

    How can the logarithm problem be solved?

    Interesting, i have not worked on logs with conjoined bases before, anyway my approach is as follows; $$p^5=x$$ and $$p^2=y$$ Let $$log_{xy}P = m$$, →$$(xy)^m = P$$ $$(P^5⋅P^2)^m = P^1$$ $$P^{7m}=P^1$$...
  18. S

    Express logarithm in another variable

    The best I can get is: $$\log_{6} 3=m-n+\log_{6} 2$$ Is it possible to get the final answer in terms of ##m## and ##n## only? If yes, I will try to do it again Thanks
  19. A

    I What is the logarithm of the derivative operator?

    I found this article which claims to have found the logarithm of derivative and even gives a formula. But I tried to verify the result by exponentiating it and failed. Additionally, folks on Stackexchange pointed out that the limit (6) in the article is found incorrectly (it does not exist)...
  20. S

    Stating logarithm in variables

    Have tried to do that but getting no result. I know ##\log \sqrt{1000} = \frac {3}{2}## . I just want to know whether it is possible to state ##\log \sqrt{1000}## in terms of u and/or v without using "weird stuff", like ##\log \sqrt{1000} = \frac{3}{2} + u - u ## (this is what I did...) Thanks
  21. OwlsInATrenchcoat

    B Logarithmic IV graphs of diodes

    Hello there, I've been working through a task (that doesn't have an answer sheet or explanation) in which we plot I against V for three different diodes. Each has a different threshold voltage and displays the usual charcteristic curve. The final question is this: "It is suggested that the...
  22. anemone

    MHB Unsolved Challenge: Natural logarithm and Exponent

    Prove $e^{-x}\le \ln(e^x-x-\ln x)$ for $x>0$.
  23. G

    How do you properly add logarithms with negative characteristics by hand?

    From the log tables: ##log(890) = 2.9494, \space log(12.34)=1.0913, \space log(0.0637)=\bar{2}.8041## I calculate by hand: ##\begin{array}{r} &2.9494\\ +&1.0913\\&\bar{2}.8041\\\hline &2.8448 \end{array}## Thus: ##log^{-1}(2.8448) \approx 699.6 \space## Which is the correct answer. Now I...
  24. W

    MHB Logarithm Identity: Prove Loga(1/x)=log1/x(a)

    If a>1, a cannot = 1, x>0, show that Loga(1/x) = log1/x(a). (COULD NOT SOLVE)
  25. J

    Trivial question on differentiating logarithms

    My Question : Shouldn't differentiating ##-log B## give ##\frac{-\delta B}{B}##? (Note : A, B and Z are variables not constants) By extension for ##Z=A^a \,B^b\, C^c## where ##c## is negative, should ##\frac{\Delta Z}Z=|a|\frac{\Delta A}A+|b|\frac{\Delta B}B-|c|\frac{\Delta C}C##?
  26. M

    A Regression analysis: logarithm or relative change?

    Hi. I am currently studying the market for equity options and the use of these to predict stock return around company earnings announcements. The dependent variable in my regression analyses have been the relative change in stock price or log-return from the day before the announcement to...
  27. B

    When can using a logarithm make solving an equation easier?

    Homework Statement Well, there is a physics problem I was solving and it is really interesting how it is officially solved. We take a small weight and hang it on a steel wire. For how much does the oscillation time change if the temperature of this wire raises for 10K? I looked up solution...
  28. G

    I Solve ln(a/b+1) Using Identity ln(a+b)=ln b + ln(a/b+1)

    How to solve "ln(a/b+1)" after applying the identity "ln(a+b)=ln b + ln(a/b+1)" ? where "ln" is natural log, a and b have variable values in them.
  29. nomadreid

    I |Li(x) - pi(x)| goes to 0 under RH?

    Extremely quick question: According to http://mathworld.wolfram.com/PrimeNumberTheorem.html, the Riemann Hypothesis is equivalent to |Li(x)-π(x)|≤ c(√x)*ln(x) for some constant c. Am I correct that then c goes to 0 as x goes to infinity? Does any expression exist (yet) for c? Thanks.
  30. hilbert2

    I Is there a name for this approximation?

    Because it holds that ##\displaystyle\int_{1}^{x}\frac{dt}{t} = \log x##, and ##\displaystyle\int_{1}^{x}\frac{dt}{t^a} = \frac{1}{a-1}\left(1-\frac{1}{x^{a-1}}\right)\hspace{20pt}##when ##a>1## it could be expected that ##\displaystyle\frac{1}{a-1}\left(1-\frac{1}{x^{a-1}}\right)...
  31. M

    MHB Logarithm properties in functional equations Show that f(1/a) = -f(a)

    Suppose that we have a function f(x) such that f(ab) = f(a)+f(b) for all rational numbers a and b. (a) Show that f(1) = 0. (b) Show that f(1/a) = -f(a). (c) Show that f(a/b) = f(a) - f(b). (d) Show that f(an) = nf(a) for every positive integer a. For (a), if ab = 1 then a = 1/b and b = 1/a. Not...
  32. R

    MHB The Limited Logarithm: Why x Can't Be <= 0

    Why x can't be less or equal to zero?
  33. DLeuPel

    I Invention of Napier’s logarithm

    I’m trying to figure out how logarithms we’re invented. In addition, what does the calculator do when I want to solve a logarithm. After researching I found out that you could compare an arithmetic progression with a geometrical one, obtaining the principal properties of exponent calculation...
  34. Philip Robotic

    Integration that leads to logarithm functions problem

    Hi everyone, So I am a high school student and I am learning calculus by myself right now (pretty new to that stuff still). Currently I am working through some problems where integration leads to logarithm functions. While doing one of the exercises I noticed one thing I don't understand. I...
  35. V

    Logarithm Function: Properties and Solutions [Attached Image]

    Homework Statement I have attached image of question.[/B]Homework Equations all the properties of log a^(logₘn)=n^(logₘa)[/B]The Attempt at a Solution in the attached image [/B]
  36. opus

    B Why is pH a negative logarithm?

    I'm going over applications of logarithms in my College Algebra class and I'm at a part where it talks about pH scales, and it shows the pH concentration of a substance to be the negative logarithm of hydronium ions. I want to know why the logarithm is negative, so I googled it and the answers...
  37. F

    Solving for time in an exponential equation

    Homework Statement In my book, there is a formula that gives the amount (in grams) of Radium in a jar after t years (100 grams were initially stored): R = 100⋅e-0.00043⋅t The book asks me to sketch the graph of the equation. I decided to find a point where the time elapsed equals the...
  38. AdrianMachin

    Ambiguity in finding errors using natural logarithm method

    Homework Statement [/B] I was reviewing this stuff and although I excelled at it once, I seem to forget some of it. For example, please consider this: Homework Equations R_C=\frac {R_1R_2} {R_1+R_2} + R_3 Here's the correct formula for its error: \Delta R_C=\frac {R_1R_2} {R_1+R_2} \left[...
  39. CalcExplorer

    A How to Convert a Complex Logarithm to a Complex Exponential

    Okay, so I'm working with a rather frustrating problem with a calculus equation. I'm trying to solve a calculus equation which I conceptualized from existing methods involving complex number fractal equations. I'm very familiar with pre-calculus, while being self-taught in portions of calculus...
  40. Rectifier

    Complex logarithm as primitive

    The problem I am trying to calculate the integral $$ \int_{\gamma} \frac{z}{z^2+4} \ dz $$ Where ## \gamma ## is the line segment from ## z=2+2i ## to ## z=-2-2i ##. The attempt I would like to solve this problem using substitution and a primitive function to ## \frac{1}{u} ##. I am not...
  41. K

    MHB A logarithm formula involving the mascheroni constant

    After watching this video: The mystery of 0.577 4k1jegU4Wb4 My problem is at position 7 mins 26 secs where he states the following: 1 - Ln = 1 1+ 1/2 - Ln2 = 0.81 1 + 1/2 + 1/3 - Ln3 = 0.73 And so on until we arrive at Eulers Mascheroni Constant Being that he is using 'Ln' have learned this...
  42. G

    Logarithm and statistical mechanics

    Hello, I'll try to get right to the point. Why and how does logarithmic dependence appear in statistical mechanics? I understand that somehow it is linked with probabilities, but I can not quite understand.
  43. alijan kk

    Understanding the Relationship Between Natural Logarithms and Their Reciprocals

    Homework Statement 1/loga(e) = loge(a) Homework EquationsThe Attempt at a Solution how they are reciprocals of each other ? is their any longer but intuative way to show this result
  44. F

    Can Discrete Logarithms Be Added in Modular Arithmetic?

    Homework Statement Let g be a primitive root for ##\mathbb{Z}/p\mathbb{Z}## where p is a prime number. b) Prove that ##\log_g(h_1h_2) = \log_g(h_1) + \log_g(h_2)## for all ##h_1, h_2 \epsilon \mathbb{Z}/p\mathbb{Z}##. Homework Equations Let x, denoted ##\log_g(h)##, be the discrete logarithm...
  45. H

    Find the derivative of this function

    Homework Statement $$y = x.\log_e {\sqrt{x}}$$ Homework Equations f(x) = g(x) h(x) f ' (x) = g ' (x) . h (x) - h ' (x) . g(x) The Attempt at a Solution $$y = x .\log_e {\sqrt {x}}$$ $$y '(x) = 1.ln \sqrt{x} + \frac{1}{2} $$ the right answer is $$ y ' = \log_{10} {\sqrt{x}} + \frac{1}{2} $$...
  46. H

    Can you help me to find the solution of logarithm equation

    Homework Statement $$\log_2 \sqrt{2x-1} < \log_4 x$$ Homework EquationsThe Attempt at a Solution $$\log_2 \sqrt{2x-1} < \log_4 x$$ $$\log_2 {2x-1}^{1/2} < \log_2 \sqrt{x}$$ $$1/2 \log_2 {(2x-1)} - \log_2 \sqrt{x} < 0$$ $$ 1/2 \log_2 {\frac{(2x-1)}{ x}} < 0$$ $$ -1/2 \log_2 {(2x-1) x} < 0$$...