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We know that ln x is increasing (Property 9), that ln 1 = 0 (Property 1), and that 1 < ln 2 < 1 (Property 11). From 2 the value y = ln 2 = 0.693 we can estimate the y-values at x = 4, 8, 16, and at x = 1 , 1 , 1 , . . . by Property 8, 2 4 8 ln 4 = 2 ln 2 1 ln = ln 2 2 ln 8 = 3 ln 2 1 ln = 2 ln 2 4 ln 16 = 4 ln 2 . . . 1 ln = 3 ln 2 . . . 8

The cascaded ampli er of Fig. 6-26(a) is built up with identical transistors for which hre hoe % 0, hfe 100; and hie 1 k . Let RE1 1 k , RC1 10 k , RE2 100 , RC2 RL 3 k , and Cc CE ! 1. Determine (a) the overall voltage-gain ratio Av vL =vs , and (b) the overall current-gain ratio Ai iL =is .

Solution: Sales Income Cost of Goods Sold: Beginning Inventory Purchases $66,000 Less: Purchase Returns 2,000 Goods Available for Sale Less: Ending Inventory Cost of Goods Sold Gross Pro t Total Expenses: Net Pro t $103,000 $30,000 $64,000 $94,000 24,000 $ 70,000 $ 33,000 27,900 $ 5,100

(a) The small-signal equivalent circuit is given in Fig. 6-26(b). From the results of Problem 6.7 with hoe 0 and RC replaced with RC2 , Av2 hfe RL RC2 100 3 103 3 103 150 hie RL RC2 1 103 3 103 3 103

2 Dx (ln x) = Dx (x 1 ) = x 2 = 1/x 2 < 0 and, therefore, the graph is concave downward. There is no horizontal asymptote (by Property 12), but the negative y-axis is a vertical asymptote (by Property 13). The graph is sketched in Fig. 34-2. Notice that ln x assumes all real numbers as values.

From the results of Problems 6.48, in which RC , RL , and RE are replaced with RC1 , hie , and RE1 , respectively, Av1 hfe RC1 hie 100 10 103 1 103 0:891 RC1 hie hfe 1 RE1 hie 11 103 100 1 1 103 1 103

Q2 RL RC2 _V (a)

Fig. 34-2

+ LL _

C2 iL hfeib2 RC2 RL + LL _

v 1 1

10

RE1 _

Fig. 6-26

make the change of variable w = ut (u xed). Then dw = u dt, and the limits of integration, t = 1 and t = v, go over into w = u and w = uv, respectively, ln v = Then, by Theorem 30.4, ln u + ln v =

Thus;

Av Av1 Av2 0:891 150 133:6

By the chain rule, 1 1 1 Dx (ln x r ) = r Dx (x r ) = r (rx r 1 ) = r = rDx (ln x) = Dx (r ln x) x x x Then, by Corollary 29.2, ln x r = r ln x + C, for some constant C. Substituting x = 1, we nd that C = 0, and the proof is complete.

(b) From the results of Problem 6.48 with RC and RL replaced with RC1 and hie , respectively, Ai1 hfe RC1 100 10 103 90:91 RC1 hie 10 103 1 103

In This :

Now, by current division at the output network, RC2 RC2 RL hfe RC2 i 100 3 103 Ai2 L 50 ib2 RC2 RL 3 103 3 103 Ai Ai1 Ai2 90:91 50 4545:4 iL hfe ib2

In the cascaded CB-CC ampli er of Fig. 6-27(a), transistor Q1 is characterized by hrb1 hob1 % 0, hib1 50 , and hfb1 0:99. The h parameters of transistor Q2 are hoc2 % 0, hrc2 1, hic2 500 , and hfc2 100. Let RL RE2 2 k ; RB1 30 k ; RB2 60 k ; R1 50 k ; R2 100 k ; RE1 5 k ; and CB Cc ! 1. Find (a) the overall voltage-gain ratio Av vL =vS and (b) the overall current-gain ratio Ai iL =iS .

34.4 Evaluate: (a) Dx (ln(x 3 2x))

(a) The small-signal equivalent circuit is shown in Fig. 6-27(b). RB R1 kR2 , Av1 From the results of Problem 6.19, with