







y=lnx+(1+ln2)/2 and y=x^2 
20190128 

From Mike: Prove that y=lnx+(1+ln2)/2 and y=x^2 touch each other.
The course is about logarithm and root functions... how should I solve this problem? Answered by Penny Nom. 





Three consecutive natural numbers 
20170611 

From Anita: Three consecutive natural numbers are such that the square of the middle number exceeds the difference of the square of the other two by 60. Find the numbers. Answered by Penny Nom. 





Differentiate ln[x(2x4)^1/2] 
20140628 

From Igwe: If y=In[x(2x4)^1/2],find dy/dx at x=3 Answered by Penny Nom. 





ln(12 x) = 6 
20110303 

From Carolyn: solve the equation
ln12x = 6
Thank you
Carolyn Answered by Penny Nom. 





A Taylor polynomial for (lnx)/x 
20100929 

From Dave: I have a series problem that I cannot solve. The problem asks for you to compute a Taylor polynomial Tn(x) for f(x) = (lnx)/x. I calculated this poly out to T5(x) and attempted to use this to identify a pattern and create a series in order to calculate Tn(x). However, the coefficients on the numerator out to F5prime(x) are as follows: 1, 3, 11, 50, 274... Ok, so the negative is an easy fix > (1)^n1. But the other coefficients are stumping me. I can't see any sort of pattern there and I've tried every trick I know. Is there another way to go about this?
Thanks! Answered by Chris Fisher. 





1/a + 1/b + 1/c < 1 
20100201 

From Mrityun: suppose a,b and c are natural numbers such that 1/a + 1/b + 1/c < 1. Prove that
1/a + 1/b + 1/c < = 41/42. Answered by Penny Nom and Claude Tardif. 





Properties of Natural Numbers 
20090724 

From nazrul: If m,n,k are natural number how can I prove that (m+n)k=mk+nk. In the proof the properties of natural number should be used. Answered by Janice Cotcher. 





Solve e^x + e^(x+1) = 250 
20080825 

From Laura: "solve this equation correct to three decimals"
e^x + e^x+1= 250 Answered by Penny Nom. 





Solving with the Natural Log 
20080728 

From CB: I am trying to solve lnx +3ln2 = ln 2/x. Every way I've tried when I go to plug in the solution it doesn't come out right. Answered by Penny Nom. 





0 = (1/t)  [ln(1+r) ln(t)] 
20071018 

From Brandi: Objective: solve for t, where r = 0.05
0 = (1/t)  [ln(1+r) ln(t)] Answered by Penny Nom. 





x= (y/z)ln((1+a^2)/(1a^2)) 
20070926 

From John: If
x= (y/z)ln((1+a^2)/(1a^2))
Please solve for 'a'
Thanks in advance! Answered by Penny Nom. 





Logarithmic differentiation 
20050523 

From Richard: I need to convince myself that I understand the process of
differentiating y=x^{x}.
The specific question is that if I have to take the logarithm of both sides
of the equation how can differentiate the following?
y= {(x+2)^{(x+2)}}/{(x+1)^{(x+1)}}  {(x+1)^{(x+1)}}/(x^{x}),
I have an idea that the differential of this fairly complex function
is itself ... am I right or wrong. Answered by Penny Nom. 





arcsech x 
20050210 

From Monica: Prove that arcsech x = ln[(1 + (1x^{2})^{(1/2)}) / x ] Answered by Penny Nom. 





39 consecutive natural numbers 
20030819 

From A student: Prove that among any 39 consecutive natural numbers it is always possible to find one whose sum of digits is divisible by 11. Answered by Penny Nom. 





Natural logarithms 
20030722 

From Amanda: I'm going into my senior year of high school. I will be taking AP calculus, and my teacher gave us some homework over the summer. However, there are two things that I do not understand how to do. The first is, she wants us to be able to generate a unit circle by hand using 30, 60 and 90 degree triangles. I have used the unit circle in trigonometry, however I was never taught how to draw it. Secondly, I need to know how to do natural logarithms without a calculator. I was not taught how to do this, and the worksheet I was given only showed me how to complete them using a calculator. Answered by Claude Tardif. 





R= 5.3lnx + x. 
20030511 

From A student: Reaction R to a dose x is given by: R= 5.3lnx + x. For a certain drug, R must not exceed 21. Show that a dose between 8 and 12 units satisfies this requirement & find, correct to 5 decimal places, the greates value of x which satisfies this condition? Answered by Harley Weston. 





Subsets of the natural numbers 
20010130 

From Christina: How do I explain why the set of natural numbers (N) cannot be equivalent to one of its finite subsets? Answered by Penny Nom. 





LOG(LN(x)) = 1 
20000728 

From An algebra student: LET F(x)=LOG X AND G(x)= LN X. SOLVE (f *G)(x)= 1 SHOW COMPOSITION AND USE DEFINITION OF LOGS. Answered by Harley Weston. 





Radioactive decay 
20000518 

From Catherine Sullivan: Please help me with the following: The radioactive isotope carbon14 is present in small quantities in all life forms, and it is constantly replenished until the organism dies, after which it decays to carbon12 at a rate proportional to the amount of C14 present, with a half life of 5730 years. Suppose C(t) is the amount of C14 at time t.  Find the value of the constant k in the differential equation: C'=kC
 In 1988 3 teams of scientists found that the Shroud of Turin, which was reputed to be the burial cloth of Jesus, contained 91% of the amount of C14 contained in freshly made cloth of the same material. How old is the Shroud according to the data?
Answered by Harley Weston. 





Two calculus problems 
20000501 

From Kaushal Shah: How Do WE Integrate the following Functions,  Integral xtanx dx
 How was natural base "e" discovered and why e=2.7.......
Answered by Claude Tardif. 





Natural numbers 
20000412 

From Sara: What is a natural number??? Answered by Penny Nom. 





Capacitor discharge rate 
20000109 

From Bill Phillips: I need to be solve for t in the following rate problem for an electrical capacitor. Vr = E(e^t/RC), e=nat. log 2.718 raised to the t/RC power. Answered by Harley Weston. 





Two derivatives 
19991116 

From Gina Renicker: The derivative of: y=e^{(xlnx)} and y=x^{2arctan(x1/2)} Answered by Harley Weston. 





Derivatives with logs 
19991026 

From Kate: What is the derivative of 5 to the 5x2 at x equals 0.8? Answered by Harley Weston. 





John Napier and e 
19990206 

From Shimin: I would like to ask about the exact definition of e, its history (like how John Napier came about discovering it) and its applications in problems and real life situations. Thank you! Answered by Chris Fisher. 





ln(x+3)+ln(x+2)=ln(x+11) 
19991124 

From Thibault Portevin: "Résolvez l'équation : ln(x+3)+ln(x+2)=ln(x+11)" Je suis bloqué, pourriezvous m'aider? Answered by Claude Tardif. 





logarithme neperien 
19991110 

From Louise Kieffer: D'où provient la valeur e ( 2,7....) des logarithmes népériens ? Answered by Claude Tardif. 

