Examples from Section 6.2
Natural Logarithms

p. 465

Properties of Logarithms

Note: In Mathematica, the natural logarithm function is denoted by Log[].

3.

a)

Log[ Sin[t] ]  -  Log[Sin[t]/5]   ==  
      Log[Sin[t]]  - ( Log[Sin[t]]  -  Log[5])  ==  Log[5]

b)

Log[ 3  x^2  -  9  x] +  Log[ 1/x ]   ==  
      Log[3 x (x  -  3)]  -  Log[x]  ==  
      Log[3 (x - 3)]  +  Log[x]   -  Log[x]  ==  
      Log[3 (x - 3)]

c)

(1/2) Log[4  r^4]  -  Log[2]  ==  
      (1/2) (Log[(2^2)]  +   Log[(r^4)])  -  Log[2]  ==  
      (1/2) (2 Log[2]  +  4 Log[r])  -  Log[2]  ==  
      2 Log[r]

Derivatives of Logarithms

15.

(∂_t) (t (((Log[t]))^2))

19.

(∂_t) (Log[t]/t)

33.

Log[(((((x^2) + 1))^5)/(Sqrt[1  -  x]))]  ==  
      5 Log[(x^2) +  1]  -  (1/2) Log[1  -  x]

(∂_x)(5 Log[(x^2) +  1]  -  (1/2) Log[1  -  x] )

Logarithmic Differentiation

39.

y  =  (Sqrt[(t/(t + 1))])

(∂_t) Log[y]  ==  (1/y) (∂_t) y  == 
      (∂_t)(Log[(Sqrt[(t/(t + 1))])])  == 
      (∂_t)((1/2) Log[t]  -  (1/2) Log[t + 1])  ==  
      (1/(2 t)) - (1/(2 (1 + t)))

(∂_t) y  ==  
      y ( (1/(2 t)) - (1/(2 (1 + t))))  == 
      (Sqrt[(t/(t + 1))]) ( (1/(2 t)) - (1/(2 (1 + t))))

50.

y = ((((x (x + 1) (x - 2))/(((x^2) + 1) 
              (2  x + 3))))^(1/3))

\(∂\_x Log[y]\  == \ \(1\/y\) ∂\_x y\  == 
      ∂\_x\((
          Log[\((\(x \((x + 1)\) \((x - 2)\)\)\/\(\((x\^2 + 1)\) 
                    \((2  x + 3)\)\))\)\^\(1/3\))\)\  == \ 
      ∂\_x\((
          \(1\/3\) 
            \((Log[x]\  + \ Log[x + 1]\  + \ Log[x - 2]\  - \ 
                Log[x\^2 + 1]\  - \ Log[2  x + 3])\))\)\ 
 == 
      1\/3\ \((1\/\(\(-2\) + x\) + 1\/x + 1\/\(1 + x\) - 2\/\(3 + 2\ x\) - 
            \(2\ x\)\/\(1 + x\^2\))\)\)

\(\(∂\_x y\  == \ 
      y  1\/3\ \((
          1\/\(\(-2\) + x\) + 1\/x + 1\/\(1 + x\) - 2\/\(3 + 2\ x\) - 
            \(2\ x\)\/\(1 + x\^2\))\)\  == \ 

      \(\@\(\(x \((x + 1)\) \((x - 2)\)\)\/\(\((x\^2 + 1)\) 
                  \((2  x + 3)\)\)\)\%3\) 1\/3\ 
        \((1\/\(\(-2\) + x\) + 1\/x + 1\/\(1 + x\) - 2\/\(3 + 2\ x\) - 
            \(2\ x\)\/\(1 + x\^2\))\)\ \)\)

Integration

57.

(∫_1^2) 2  Log[x]/x \[DifferentialD]x  ==  
      (∫_1^2) 2  Log[x] \[DifferentialD]Log[x]  ==  
      (Subscript[((Log[x]^2)), 1]^(2 )) == Log[2]^2

(∫_1^2) 2  Log[x]/x \[DifferentialD]x

64.

\(∫\_\(π/4\)\%\(π/2\)Cot[t] \[DifferentialD]t\  == \ 
        ∫\_\(π/4\)\%\(π/2\)\(Cos[t]\/Sin[t]\) 
            \[DifferentialD]t == 
        ∫\_\(π/4\)\%\(π/2\)\(1\/Sin[t]\) 
            \[DifferentialD]Sin[t]\  == 
	
        \((Log[Sin[t]])\)\_\(π/4\)\%\(π/2\)\  == \ 	
        Log[Sin[π/2]]\  - \ Log[Sin[π/4]]\  == \ 
        0\  - \ Log[\@2\/2]\  = \ \(1\/2\) Log[2]\)

(∫_(π/4)^(π/2))Cot[t] \[DifferentialD]t

Theory and Applications

69.

b) Find the extreme values of cos (ln (x))) on [-1/2, 2].

Plot[Cos[Log[x]],  {x, 1/2, 2}]

Solve[(∂_x) Cos[Log[x]]  ==  0,  x]

(∂_x) Cos[Log[x]]  ==  0

Log[x]  ==  0;  x  ==  1;

Cos[Log[1]]

Cos[Log[1/2]]  // N

Cos[Log[2]]  // N

cos (ln (2)) =  cos (ln (1/2))since ln (1/2)  =  -ln (2) and cos (-x)  =  cos (x).

The maximum of cos (ln (x)) is 1 at x  =  1 and the minimum is cos (ln (2))  =  0.769239 at x  =  2.

Grapher Explorations

87.

Plot[{Log[x], Log[2 x], Log[4 x], Log[8 x], Log[16 x], 
        Log[32 x]}, {x, 0.001, 10}]

The plots all seem to be the same as ln (x), shifted vertically. This behavior follows from the fact that ln ((2^n) x)  =  n ln (2)  +  ln (x), so each successive plot is shifted up by ln (2)  ≅  0.69315.

88.

Plot[Log[Abs[Sin[x]] ],  {x, 0, 22}, PlotRange -> {-2, 0}]

The graph is periodic because the inner function sin (x) is periodic. The values of  | sin (x) |  run from 0 to 1 as x runs from 0 to π/2 and then from 1 down to 0 as x runs from π/2 to π, repeating this pattern over every interval of length π. So ln ( | sin (x) | ) runs from -∞ to 0 and back down to -∞ over the same intevals.