Is lnn\ln n transcendental for all rational n>1n>1?

I know that lnn is transcendental for all integer n>1. But does this still hold for non-integer rational values of n>1? For example, is ln1.5 transcendental? EDIT: Somehow managed to overlook the fact that cases like lne=1 are not transcendental 😛 The question has been revised to only include rational numbers n. Answer No, what … Read more

Prove that ln(x)ln(x) is concave by using the following definition for ln(x)ln(x)

Using the following definition: ln(x)=∫x11tdt Show that ln is concave. So basically what I need to show is that for x,y∈R+,x≠y and for some t∈(0,1):ln(tx+(1−t)y)<tln(x)+(1−t)ln(y). We have ln(tx+(1−t)y)=∫tx+(1−t)y11tdt=∫tx11tdt+∫(1−t)ytx1tdt Assuming that F is the integral of 1tdt we get: F(tx)−F(1)+F((1−t)y)−F(tx)=F((1−t)y)−F(1) but this doesn’t lead my anywhere. Any hints please? Answer If F′ is decreasing then F … Read more

The minimum value of log10x+logx10\log_{10}x+\log_x 10

Notation: log:=log10 logx+logx10 =logx+1logx =log(x⋅1x) =log1 =0 Is the process correct? I doubt this is wrong. Please help. Thanks. Answer Note that logab=lnblna. Hence you want ot minimize y+1y with y=lnxln10. “Clearly”, this minimum is 2 and achieved when y=1, i.e., when lnx=ln10 and finally x=10. Why y=1y≥2 for y>0? Well, we have y−2+1y=(√y−1√y)2≥0 with … Read more

Solving for xx in an exponential equation

Say we the following equation F(x)=exp(a+bx)1+exp(a+bx) Now we set x=0 and we want to solve for a as a function of F0. So that, we have: F0=exp(a)1+exp(a) Can someone please walk me though the logarithmic transformation on how we achieve this end result a=ln(F01−F0) Answer F0=ea1+ea F0+ea⋅F0=ea F0=ea−ea⋅F0 F0=(1−F0)⋅ea ea=F01−F0 a=ln(F01−F0) AttributionSource : Link , … Read more

$c$ is the value of $x^3+3x-14$ where $x=\sqrt[3]{7+5\sqrt2}-\frac{1}{\sqrt[3]{7+5\sqrt2}}$.Find the value of $a+b+c$

$a=\sqrt{57+40\sqrt2}-\sqrt{57-40\sqrt2}$ and $b=\sqrt{25^{\frac{1}{\log_85}}+49^{\frac{1}{\log_67}}}$ and $c$ is the value of $x^3+3x-14$ where $x=\sqrt[3]{7+5\sqrt2}-\frac{1}{\sqrt[3]{7+5\sqrt2}}$.Find the value of $a+b+c$. I tried to solve and simplify this problem but no luck.Please help me.Thanks. Answer $$a^2=57+40\sqrt2+57-40\sqrt2-2\sqrt{57^2-40^2\cdot2}=2\cdot57-2\sqrt49=100$$ $$a=10$$ $$b=\sqrt{5^{2\log_5 8}+7^{2\log_7 6}}=\sqrt{8^2+6^2}=\sqrt100=10$$ $$x=a-\frac1a,\quad a=(7+5\sqrt2)^{\frac13}$$ $$x^3+3x-14=\left(a^3-3a+\frac3a-\frac1{a^3}\right)+3\left(a-\frac1a\right)-14\\ =a^3-\frac1{a^3}-14=7+5\sqrt2-\frac1{7+5\sqrt2}-14=7+5\sqrt2-\frac{7-5\sqrt2}{-1}-14\\ =0=c$$ AttributionSource : Link , Question Author : Vinod Kumar Punia , Answer Author : Kay … Read more

What is the nthn^{th} derivative of logx(e)\log_x(e)?

I happened to stumble upon the integral of logx(e), finding it to apparently be non-elementary. So I had to see if I could discern a pattern by differentiating, much like finding the integral of W(x). f(x)=logx(e)=1ln(x)f′(x)=−1xln(x)2f′′(x)=1(lnx)2+12×2(lnx)3 Hopefully, there is a pattern that can be used to find f(n). So please find f(n) and if we … Read more

Rejecting a solution.

Why does it for x2=9 we get two solutions, while if we use the “log both sides” property the negative solution is rejected? which method is true and why? Answer The property log(an)=nlog(a) is true if a>0 and n∈N, but not if a<0, since log is only defined on positive real numbers. Therefore, using log … Read more

How to express log52\log_5 2 in terms of a and b (Refer to qn)

In my textbook, I came across this interesting question which I am currently struggling to solve: If log62=a and log53=b, express log52 in terms of a and b The solution given is ab1−a but I do not know the working behind this. What is it? Answer An alternative, although I agree with the exponential approach, … Read more