Logarithm Definition and 339 Threads

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 $$ y ' = \log_{10} {\sqrt{x}} + \frac{1}{2} $$...

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

Homework Statement [/B] Given a a log-log graph with 8 lines, you must determine the equation of each line in its original relationship. The slope of the graph (m) gives the power of the original relationship. Examples: if m = 2, 3; then y ∝ x^2, x^3, etc. if m = -1, -2; then y ∝ 1/x...

I have a couple of college algebra textbooks , but none has some proper intro to logarithm basics . I like to keep a book for logarithm alone , i see some online PDF's on logarithm , but all are sort of messed up . Please help

Homework Statement Homework Equations log_2 x = y 2^y = x 3^2^y The Attempt at a Solution log_2 x = y 2^y = x log_2 {log _3 {log _2 { log_3 {2^y} } } } what am I suppose to do?

Dear community, I stumbled upon this ecology article (https://www.witpress.com/elibrary/dne/4/2/402, page 4) and have some confusion about a statement in there: "Before further unpacking the formal defnition of entropy, one would be justifed in asking why not simply choose (1 – p) instead of...

Homework Statement Homework Equations $$ \log_{x} a \ means \ 10^x = a $$ $$ \log_{x} {a^n} = n \log_{x}{a} $$ The Attempt at a Solution $$ \log_{10} {ax} \log_{10}{bx} + 1 = 0 $$ $$ \ means \log_{10} {ax} = 1 \log_{10}{bx} = -1 \ or \ vice \ versa $$ $$ \ for \log_{10} {ax} = 1 \ so \ ax...

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 =...

Hi I wonder how I can rewrite log-x to log -x^2

I tried with Google but I couldn't find anything, so here goes: When I "use ln on a quantity" (I don't really know how to phrase it in english, as we just have a verb for it), say, I have n = 0.00149 kg/m*s, and I put it into the ln, so now I have ln(0.00149 kg/m*s) what happens to the SI Units...

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...

Homework Statement 2 - log10 3x = log10(x/12) Homework Equations logab=b log a log(a/b)= log a - log b The Attempt at a Solution 2 + log10 12= log10 x - x log10 3 Start seems simple but cannot see where to go from here, taking exponentials doesn't seem to help. Not sure what the next steps...

Homework Statement Intensity of sound is (W/m^2) and is inversely proportional to the the square of the distance measured from the sound source. The noise level of a risiing jet aircraft at the distance of 30m, is 140 dB. How far from the jet aircraft is the noise level at the level of 120 dB...

(This is absolutely not a HW problem) I had posted in a much older thread, so just wanted to post again in a new one. How would you solve, for x: 5x = 4x + 1 x = 1 is clearly an answer, but I get struck when trying to solve using logs. If I take log5 of both sides I get, x = log5 ( 4x+ 1)...

Today I came across a high school math book which has a particular problem in the logarithms chapter. It has $$ \log_{10}{0.2913} = -1.4643 $$ Trying to verify it with a calculator, I get -0.53566. There's a log table attached at the end which agrees with the calculation made in the book. To...

Could someone please explain to me why this equation holds: alog x = blog x / blog a For all the all other rules concerning logarithms I could derive them from the rules for exponentiation, except for this rule. Could someone please explain to me how this derivation works? And how the value of...

If you had something particularly nasty like ##log_{10}(9)## = ? I asked my teacher about how to begin approaching that kind of computation of logarithm. He was not very interesyed in explaining the procedures of estimating the value of the log. Perhaps the procedure went beyond the scope of...

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 = ...

Hello I have three limits to calculate, based on a given limits. What I know is: \[\lim_{x\rightarrow 0}\frac{ln(1+x)}{x}=1\] And based on this, I need to find (without L'Hopital rule), the following: \[\lim_{x\rightarrow 0}\frac{ln(1-x)}{x}\] \[\lim_{x\rightarrow 0}\frac{ln(1+x^{2})}{x}\]...

Negative number multiplied by itself an even number of times gives us a positive number. Why does log to -10 base of 100 not equal 2? thanks in advance.

Homework Statement Calculate the integral: ## \int_{a}^{b} \frac{1}{x} dx ## Homework Equations - The Attempt at a Solution In high school we learned that: ## \int_{a}^{b} \frac{1}{x} dx = ln(|x|) + C ## because the logarithm of a negative number is undefined. However, in my current maths...

½Logb(x2-1)≈logb(x) This is an easy and useful way to calculate the log of any natural number, including primes, it won't ever give a precise result, obviously (because of the -1), but as "x2-1" will always have divisors smaller than "x", you can easily calculate the approximation by using the...

I'm confused on this question. The equation m log p (n) = q can be written in exponential form as.. The answer on the work sheet is p^(q/m)=n but shouldn't it be P^(qm) = n ? According to the power rule? My teacher explained this by writing down for me log p (n) = q / m but I'm confused here

I came across a guy claiming that the "best approximation" for the natural logarithm of a number is this: ln x=2^n*(x^(2^-n)-1) Oddly enough, it seems to work rather well! I don't really get why it does... I also don't know if it has a limit, I couldn't test it as I don't have access to my...

Hello. Let's have any non-zero complex number z = reiθ (r > 0) and natural log ln applies to z. ln(z) = ln(r) + iθ. In fact, there is an infinite number of values of θ satistying z = reiθ such as θ = Θ + 2πn where n is any integer and Θ is the value of θ satisfying z = reiθ in a domain of -π <...

Hello every one. Today's question is: if I'm writte a graph using logarithm scale, must i use the dimensions of the graph "logarithmized" too ? i.e : A Distance x Force graph must have its dimensions as log (m) x log (N) ? or it is just valid for the module ? Thanks for the tips

hi guys.. i need help to solve logarithm problem how to find x? thanks any help..

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...

I have a derivative of a function with respect to ##\log \left(r\right)##: \begin{equation*} \frac{dN\left(r\right)}{d \log\left(r\right)} = \frac{N}{\sqrt{2\pi} \log\left(\sigma\right)} \exp\left\{-\frac{\left[\log \left(r\right) - \log\left(r_M\right)\right]^2}{2...

The problem Solve ## e^x-e^{-x} = 6 ## . The attempt $$ e^x-e^{-x} = 6 \\ e^x(1-e^{-1}) = 6 \\ e^x = \frac{6}{(1-e^{-1})} \\ x = \ln \left( \frac{6}{1-e^{-1}} \right) \\ $$ The answer in the book is ## \ln(3 + \sqrt{10})## Could someone help me?

I'm trying to use the saddle point method to solve the following integral: Z = (1/sqrt{2 pi t}) ∫_{1}^{infinity} ds (1/sqrt{2 pi s}) exp{ p [-s ln(s/t) +s] } cos(2 pi L~ p ~ s), as p → infinity Mod edit to make integral more readable: $$Z = \frac{1}{\sqrt{2\pi t}} \int_1^{\infty} \frac 1...

1. I hit this logarithm mathematics problem.2. (1/3)^x = log<a>x For clarifications, <a> means the base a 3. I have used GeoGeBra to graph them and managed to find the intersection. But is there a solution for the exact value(s) of x?

Homework Statement If a,b,c are positive real numbers such that ##{loga}/(b-c) = {logb}/(c-a)={logc}/(a-b)## then prove that (a) ##a^{b+c} + b^{c+a} + c^{a+b} >= 3## (b) ##a^a + b^b + c^c >=3## Homework Equations A.M ##>=## G.M The Attempt at a Solution Using the above inequation, I am able...

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)##...

Let the reals $a, b, c∈(1,\,∞)$ with $a + b + c = 9$. Prove the following inequality holds: $\sqrt{(\log_3a^b +\log_3a^c)}+\sqrt{(\log_3b^c +\log_3b^a)}+\sqrt{(\log_3c^a +\log_3c^b)}\le 3\sqrt{6}$.

Already tried everything, mantissa, exponent, just do not know how to solve the problem, I would love to be able to understand it! 5 (1-e ^ -0.212765957) = e = 2.71 ^ = -0.212765957 here is my biggest question!

Log(0.0058) is ( -2.2365720064), its characteristic or integral part is (-2) but not (-3). As per rules of logarithm, Its characteristic or integral part must be (-3 ) because of two zeros plus 1 (as per rule) but its characteristic is (-2), similiarly log(0.0648) = -1.188424249941 but...

Prove that, for all real $a,\,b,\,c$ such that $a+b+c=3$, the following inequality holds: $\log_3(1+a+b)\log_3(1+b+c)\log_3(1+c+a)\le 1$

Homework Statement Solve ##y=\mathrm{exp}(\frac{-x\pi}{\sqrt{1-x^2}})## for x when y = 0.1 Homework Equations ##\mathrm{ln}(e^x)=x## The Attempt at a Solution ##\mathrm{ln}(0.1)=\frac{-x\pi}{\sqrt{1-x^2}}## ##(\frac{-\mathrm{ln}(0.1)}{\pi})^2=\frac{x^2}{1-x^2}##

So, I found this method, I don't think I was the first to, though, but I don't see any post related to this anywhere on the internet, so maybe there's a slim chance I was the first? Anyway, it doesn't really matter. The method does not give the precise result, only approximations, but I find it...

5X = (1+X)^15 What to do with this kinds of problems? How do I solve for X? I encounter this problems on engineering economy(Annuities)... I tried doing logarithms but no success.

" Solve the equation: 2e^(-x) = 3e^(0.1x) " I've been fiddling around with this and I have no idea what I'm supposed to do. I know the final answer should be something like: x = [(2/3)log e] / 1,1 The only step I've managed to do was: (2/3) x e^(-x) = e^(0.1x) But after that, I don't know...

I need to prove: (n+1)*(log(n+1)-log(n) > 1 for all n > 0. I have tried exponentiating it and I got ( (n+1)/n )^(n+1) < e. And from there I couldn't go any farther, but I do know that it is true by just looking at its graph. Could anybody help me please?

I know that if you have x states then you need log2(x) bits to encode them. For example a coin has 2 states and you need 1 bit which is log2(2). It also works for numbers between 0 and 1 for example if you halve the amount of states you need to add log2(1/2) bits which is -1. So what does...

Temperature of a system is defined as $$\left( \frac{\partial \ln(\Omega)}{ \partial E} \right)_{N, X_i} = \frac{1}{kT}$$Where Ω is the number of all accessible states (ways) for the system. Ω can only take discrete values. What does this mean from a mathematical perspective? Many people say we...

I am interested in track & have found through some hard work that for 800 that 1'42.00 is same value as for 3'44.00 for 1600 ( note : for 1600m not 1 mile ) It is a Log relationship but I can't quite settle on what the value for equivalence should be for 1500m. I have a provisional figure of...

Homework Statement log(cosx)sinx = 4*log(sinx)cosx Homework Equations 3. The Attempt at a Solution i tried to solve it and uploaded my work but my last part reads (cosx)^2 = sinx (cosx)^(-2) = sinx its weird that i don't see where i have made a mistake[/B]

(3x)^{1+\log_3(x)}=3 How do I solve this one?

How do i solve Log_a (100) if Log_a (2) = 20 and Log_a (5) = 30 I got to 2^(1/20) = 100^(1/x) and 5^(1/30) = 100^(1/x) but didnt know how to go any further.

Hi guys, new here. Thank you for reading my post. I'm posting today because although I'm an engineer, I have some brain tumors that have impeded my ability to process most math above basic algebra (despite the fact that prior I'd gone all the way through stats in college). It didn't really start...