Explanation: This integral may seem tricky at first, but it comes apart quite quickly once you realize a certain u-substitution. If we let #u=ln (ln (x))#, we get by the chain rule that the derivative is: # …

M(t)=M_0\\exp(-kt); \\qquad k=\\ln(2)/t_{1/2}; t = 1/k\\ln(M_0/M) = t_{1/2}/\\ln(2)\\ln(M_0/(0.1M_0)) = [\\ln(10)/\\ln(2)]t_{1/2} = 3.322\\timest_{1/2} = 16, 610 Yrs Let M_0 be the mass of carbon-12 at the …

Taking logs of both sides, # lne^x=ln [1/sqrt [e-1]]# #but ln [e^x]=x# and so #x=ln [1/sqrt [e-1]]#. Hope this helps. Answer link

Rule 1: sqrt {"nonnegative"} 7-ln (x-1) ge0 by adding ln (x-1), => 7 ge ln (x-1) by raising e to both sides, => e^7 ge e^ {ln (x-1)}=x-1 by adding 1, => e^7+1 ge x Rule2: ln ("positive") x-1 > 0 by adding 1, => x …

#1/20 = (1/2)^ (t/ (3.8" days"))# Use the natural logarithm on both sides: #ln (1/20) = ln ( (1/2)^ (t/ (3.8" days")))# Use the identity, #log_b (a^c) = (c)log_b (a)#, on the right side of the equation: #ln (1/20) = …

E (n) = (sum_ (i=1)^n 1/i) / n E (5) = 0.45dot (6) E (100) ~~ 0.05187 E (n) = (sum_ (i=1)^n 1/i) / n = (H (n))/n where H (n) is the nth Harmonic number. As n -> oo, H (n) - ln (n) -> gamma, where gamma is …

Differentiate both sides: # (d (ln (y)))/dx = (d (sqrt (cos (x)))ln (sin (x)))/dx# Use the chain rule on the left: #1/ydy/dx = (d (sqrt (cos (x)))ln (sin (x)))/dx# Use the product rule on the right: #1/ydy/dx = (d (sqrt …

Nov 24, 2015 · f (x) = ln (x^2-1)^ (1/2) First, let's remember the following logarithmic rule: ln x ^a = a * ln x According to this rule, your function can be simplified as follows: f (x) = 1/2 ln (x^2 - 1) Now, let's …

What is the nth term of the sequence # ln (2/1),ln (3/2),ln (4/3),.. #? What is the limit as #n rarr oo# Calculus

f^ (') (x) = (cos (x)ln (x)+sin (x)/x)/2 The function given is: =>f (x) = ln (sqrt (x^ (sin (x)))) Let's first simplify sqrt (x^sin (x)). Using the law: u (x) = e^ln (u (x)) " and " ln (a^b) =bln (a) we get: =>sqrt (x^sin (x))= …