Logarithm Definition and 339 Threads

it is correct to say that if we consider the whole of R as the domain, the function ln(x)is not continuous, whereas if we consider the domain of the function as the domain, then it is continuous?

Markscheme; Not many international students would understand single logarithm as expected by examiners. In my take ##x = \dfrac{\ln 2}{2}## is single logarithm and therefore a full ##4## marks ought to be awarded. In any case, the form; ## \dfrac{\ln 2}{2} = \ln 2^\frac{1}{2}## are one...

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...

##\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##...

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 ##...

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...

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##...

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.

##\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...

##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##...

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?

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

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...

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 &=...

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...

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...

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} =...

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...

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 $$log_{xy}P = m$$, →$$(xy)^m = P$$ $$(P^5⋅P^2)^m = P^1$$ $$P^{7m}=P^1$$...

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

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)...

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

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...

this is my working...

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

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...

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

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##?

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...

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...

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.

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.

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)...

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...

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

What is error in the picture?

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...

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...

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]

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...

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...

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[...

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...

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...

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...

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.

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

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...