# What is Logarithms: Definition and 257 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:

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. ### Solve the given problem involving logarithms

In my working i have, ##\dfrac{\log_{11} x }{\log_{11} 4}= \log_{11} (x+6)## ##\dfrac{\log_{11} x }{0.5781}= \log_{11} (x+6)## ##\log_{11} x = \log_{11} \left[(x+6)\right]^{0.5781}## ##x^{1.729} = x+ 6## ##x^{1.729} -x-6=0## Having ##f(x) = x^{1.729} -x-6## At this point i made use of...
2. ### B Integral of cosecant function: understanding different approaches

Hi, PF Trigonometric Integrals "The method of substitution is often useful for evaluating trigonometric integrals" (Calculus, R. Adams and Christopher Essex, 7th ed) Integral of cosecant...
3. ### B Video: Mathologer on visual logarithms (and hyperbolic trig)

Mathologer (https://en.wikipedia.org/wiki/Burkard_Polster) has a nice video using known (but not well-known) geometric motivations of the natural logarithm and the hyperbolic functions... and he makes brief mentions of special relativity I've been using similar motivations to support geometric...
4. ### How many positive integers x satisfy this logarithmic inequality?

The whole solution is a bit long, which I'll attach but the part I'm stuck at is, assuming everything else above it is correct, is 4 < (log x - 3)(8-log x) Note ; inequalities aren't technically taught yet in the course, so please try to make the solution not go too deep into that. If that...
5. ### Solve the given problem involving logarithms

Part (a) ##s=ab^t## ##\log s= \log a+ t\log b## Expression on the right hand side increases linearly with ##t##Part(b) ##s=120 ×1.15^t## ##\log s = \log 120+t\log 1.15## ##\log s =2.08+0.06t## From graph, y-intercept = ##2.08## ##m=\dfrac{2.45-2.08}{6-0}=0.06## Part (c)...
6. ### Rearranging formula using logarithms

My attempt that doesn’t work. t=-RC ln(1-I/6.7)
7. ### Is there a rule that states that I should not divide in this scenario?

So basically this is how I solved this problem: 1. ##f(x)=\log _{2} x^2 - 1## 2. ##0=\log _{2} x^2 -1 ## 3. ##1= 2\times \log _{2} x## 4. ##\frac{1}{2}= \log _{2} x## 5. ##2^{\frac{1}{2}}=x=\sqrt{2}## So I wrote coordinates to be (##\sqrt{2}##, 0) But apparently, that is not the only solution...

14. ### I Transforming an equation with logarithms

I want to approximate the logarithm of the Binomial coefficient log (n!/ ((n - m)! m!) with the the Stirling approximation log x! ≈ x log x - x I got n log n - m log m - (n - m) log(n - m) but I want (n - m) log (n/(n - m)) + m log (n/m) Can someone help how to transform the first...
15. ### A problem involving logarithms

This is my work. I don't know whether this is correct or not. What do you think of the problem. Thanks!
16. ### I Thermodynamics Differentiating logarithms

<mentor: change title> In thermodynamics, there is a function which, for the three variables x, y, and z, can be given as ##G = xG_x+yG_y+zG_z + N[x\ln(x) + y\ln(y)+ z\ln(z)]+E(x,y,z)## where G_x, G_y, G_z, and N are some constants and E is some arbitrarily complicated term. There is a...
17. ### Mathematical methods: logarithms question

I know that sinhx = 1/2(e^x-e^-x) and that e^2x-1 = e^x(e^x-e^-x) and similar identities but don't know how to get any further. Any hints at where to go with this would be appreciated.
18. ### Proof with recursion and logarithms

Homework Statement: Suppose f(n) is a function that accepts an integer n as a parameter. When called with n = 1, it executes 2 instructions. When called with a larger value of n, it executes 10n + 30 instructions, two of which are f(n/2). Prove that f(n) executes 10n lg n + 32n − 30...
19. ### 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##?
20. ### A What is the significance of large logarithms in RG calculations?

Hello! I am reading Schwarz book on QFT and I am at the renormalization group part. I understand the math and the fact that grouping terms perturbatively and requiring them to be 0, when taking the derivatives with respect to a given arbitrary chosen scale, allows perturbative corrections to the...
21. ### B Issue With Algebra of Logarithms

I was just doing a homework problem that involved logarithms. I noticed that order of operations matters when applying logarithm rules. I'll use a different example from my homework problem to illustrate what I'm talking about. ln(5*2^3) does not equal 3*(ln(5*2)) Apparently you have to do...
22. ### MHB Solving for an exponential equation using logarithms 16^{x}-5(4)^{x}-6=0

Hello I'm having trouble solving for this exponential equation : 16^{x}-(5,4)^{x}-6=0 I used some logarithms properties but can't get anything close to the following solutions here I tried using log base 16 : log16(16^{x})-6=log16((5,4)^{x}) ; then x - xlog16(5,4)=6 ; factorizing x ...
23. ### MHB Which of the two logarithms is larger: log_7 (10) or log_11 (13)

Without using a calculator, prove which of the following is larger: \log_7 10 or \log_{11} 13
24. ### MHB Solve Logarithms Problems: Find Profit & Interest Rate

Hello all, I have a few small questions regarding logarithms, which I would like to ask your help with. 1) A car loses 50% of it's value every 4 years. How many years does it take for the car to lose 1/3 of it's value ? (I think you need log with basis of 2 here, but not sure) 2) A company...
25. ### Using logarithms in vector Calculus

Homework Statement My mentor has run me through the derivation of equation (3) bellow. I am unsure how he went from (1) to (3) by incorporating the log term from eq(2). In eq(3) it seems he just canceled the relevant n terms and then identified 1/n as the derivative of L however if this were...
26. ### B Derivation with logarithms and product

I went through an example question that showed me how to solve the question but I'm not sure if I've misunderstood something or if they did a mistake. Question: Derivate y = (1/ax)ax ln(y) = ln( (1/ax)ax ) = ax( ln(1) - ln(ax) ) = -ax ln(ax) (1/y)(dy/dx) = -ax * ax ln(a) - a * ln(ax) dy/dx =...
27. ### Exponential and logarithmic Equation Problems

Homework Statement Evaluate each of the following expressions without using a calculator. 1) log216√8Solve for the unknown value in each of the following equations without using a calculator. 2) 3(x+4)−5(3x)=684 3) 7(42x)=28(4x) Homework Equations Exponent law for multiplication The...
28. ### I Derivative of 4^x: My Exam & Answer Explained

On my exam, we had to find the derivative of 4^x. This is what I did Y=4^x lny=xln4 y=e^xln4 and then finding the derivative for that I got, (xe^(xln4))/4 My professor said that it was wrong and even after I told her what I did to get the answer. She told me the answer was (4^x)ln4 . Which I...
29. ### Multiple exponents for logarithms

Homework Statement 2^(t-1) * 6^(t-2) = 20,000 Homework EquationsThe Attempt at a Solution I have no idea how to solve this, although I do understand the basics of logarithms
30. ### B Find ##f(x)## such that f(f(x))=##log_ax##

I was thinking about extending the definition of superlogarithms. I think maybe that problem can be solved if we find a function ##f## such that ##fof(x)=log_ax##. Is there some way to find such a function? Maybe the taylor series could be of some help. Or is there some method to find a...
31. ### Logarithm question using compound interest formula

Homework Statement A colony of ants will grow by 12% per month. If the colony originally contains 2000 ants how long will it take for the colony to double in size? Answer - 6.12 months Homework Equations A = P(1+r/n)nt The Attempt at a Solution r = 12% = 0.12 n = 12 P = 2000 A = 4000 t = ...
32. ### Mastering Logarithms: Simplifying Complex Expressions with Multiple Logs

Homework Statement can we put 3log2(x)-4log(y)+log2(5) in one logarithm it try in all the ways but i can't find the solution . Homework Equations loga(b)=logx(b)/logx(a) log(b*a)=log(b)+log(a) The Attempt at a Solution log2(5x^3)-log(y^4) log2(5x^3)-log2(y^4)/log2(10)
33. ### Using Properties of Logarithms

Homework Statement [/B] Question: Use the properties of logarithms to rewrite and simplify the logarithmic expression. 1. log_{5}\dfrac{1}{250} Homework Equations Product Property: log_{a}(uv) = log_{a}u + log_{a}v Quotient Property: log_{a}\dfrac{u}{v} = log_{a}u - log_{a}v Power Property...
34. ### I Why is b>1 and x,y positive in logarithm definition?

A simple doubt came to my mind while browsing through logarithmic functions and natural logarithms we define $$\log_b(xy) = \log_b(x) + \log_b(y)$$ Here why is the condition imposed that b>1 and b is not equal to zero and that x and y are positive numbers? Is it something to do with the...
35. ### Understanding Integration by Parts: Exploring the Formula and Solving Examples

|3^xlog3dxI don't even know where to start. I know that the formula is |u.dv = uv - |v.du u=3^x v=log3
36. ### Proving 1/3 < log_{34} 5 < 1/2

Homework Statement Prove that ##{1/3} < log_{34} 5 < {1/ 2}## Homework Equations ##log_b a = {1/ log_a b}## ##logmn = logm + logn##The Attempt at a Solution ##log_{34} 5 = {1/ log_5 34}## ##= 1/(log_5 17 + log_5 2)## ##=1/(1 + log_5 3 + log_5 2 + something)##...
37. ### Solving an equation with logarithms

can somebody help me solving: A*ln(1+A*x)-2*ln(x)-B=0 I thank you in advance

39. ### MHB How to Solve for x in a Logarithmic Equation?

The question is \log_{x}\left({10}\right)+log(x)=2 where (obviously) I have to find x. I tried changing the base, $\frac{log(10)}{log(x)}+log(x)=2$ $\frac{1}{log(x)}+log(x)=2$ ${log(x)}^{-1}+log(x)=log(100)$ but I could go no further. Whatever I try, I always got a wrong answer. By guessing...
40. ### MHB Help with Logarithms: Find 2^A

So, there's this problem: A = \frac{1}{6}((\log_{2}\left({3})\right)^3 - (\log_{2}\left({6})\right)^3 - (\log_{2}\left({12})\right)^3 - (\log_{2}\left({24})\right)^3) Find 2^A What I've figured out is that all the logs factorize to 3 + 2 to some power of n...
41. ### Easy and useful way to calculate Log(a+b)

a>b ⇒ Lim(a–b)→0 Logc(a+b) ≈ Logc(a) + Logc(√a⋅b) b>a ⇒ Lim(b–a)→0 Logc(a+b) ≈ Logc(a) + Logc(√a⋅b) a∈ℝ b∈ℝ c∈ℝ / c>0
42. ### Is My Solution to This Log Question Correct?

Hi folks, I'm revisiting logs for the first time in a long time through distance education and I was wondering if someone could have a look over a question I've answered and let me know if I've done it correctly or if I'm way off please Find x if Log3(10x – 1) – 2 = 2log3x I instantly divide...
43. ### Can Logarithmic Properties Simplify This Equation?

Homework Statement For clarification a have posted the equation below as a picture file.[/B] Homework Equations log(a*b)= loga + logb log(a/b) = loga-logb log(a^n) = nloga[/B] The Attempt at a Solution I don't know how to start. I can't remember the rule for powering the logarithms if there...
44. ### Complex Integrals, Antiderivatives, Logarithms

I've been teaching myself a little bit of Complex Variables this semester, and I had a question concerning complex integrals. If I understand correctly, then if a function f has an antiderivative F , then the line integral \int_C f(z) dz is path independent and always evaluates to F(z_1)...
45. ### Solve for x: 2^3+2^3+2^3+2^3=2^x

Homework Statement Solve the following equation: 2^3+2^3+2^3+2^3=2^x[/B]Homework Equations log(a)^x=x*log(a)[/B]The Attempt at a Solution What i attempted was to log both sides, bring down the exponents, and summarize them. This left me with 12*log(2)=x*log(2). I then divide both sides by...
46. ### Is It Possible to Prove Logarithms Using Equations?

Homework Statement Proof xlogby=ylogbx Homework Equations logb is a one to one function, that is, if logbx=logby hint take the log in both sides of the equation and use the previous hint The Attempt at a Solution xlogby=ylogbx logby log x = logbx logy logby / logbx=logy/log x...
47. ### Can I Use Logarithms to Solve Exponential Equations?

So, recently I have been learning about logarithms and how to solve exponential equations with the help of logarithms, but I am curious if I can take the log of both sides of an equation like this in order to solve it? If not, then could someone explain why? Thanks.
48. ### Checking for Accuracy in homework

I noticed the scan was cut off on the second image at the bottom right, but I came up with x= 31/5 My first test in Calc I begins tomorrow and I want to know that I'm headed in the right direction. I think I understand to some extent how logarithms can be expanded and condensed though I'm...
49. ### Why Does $$\ln \frac{(x+1)}{(x-1)} \geq 0$$ Imply $$x > 1$$?

Homework Statement \ln \frac{(x+1)}{(x-1)} \geq 0 Homework Equations \ln \frac {a}{b} = \ln a - \ln b \ln \frac {(x+1)}{(x-1)} = \ln (x+1) - \ln (x-1) The Attempt at a Solution \ln (x+1) \geq \ln (x-1) e^{\ln (x+1)} \geq e^{\ln (x-1)} (x+1) \geq (x-1) According to Wolfram, the...
50. ### Logarithm problem that I'm stuck on

The problem statement ln(x) = 5 -x Solve for x. The attempt at a solution ln x = 5 - x \\ e^{ln x} = e^{5 - x} \\ e^{ln x} = \frac{e^5}{e^x} \\ x e^x = e^5 Here is the place where I get stuck.