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Department of Mathematics, University of Wisconsin-Madison
Math 114
Worksheet Sections 5.4-5.
- Change each exponential statement to an equivalent statement invoking a logarithm.
(a) 16 = 4^2
Solution: log 4 16 = 2
(b) ex^ = 8
Solution: ln 8 = x
- Change each logarithmic statement to an equivalent statement involving an exponent.
(a) log 3
Solution: 3 โ^2 = (^19)
(b) ln x = 4
Solution: e^4 = x
- Using the given function f , (a) find the domain of f , (b) graph f , (c) from the graph, determine the range and any asymptotes of f , (d) find f โ^1 , (e) find the domain and the range of f โ^1 , (f) graph f โ^1. (a) f (x) = log(x โ 4) + 2
Solution: a) The domain of f consists all x for which x โ 4 > 0, that is x > 4. Domain of f : (4, +โ) b)
o (^) x
y
c) Range of f : (โโ, +โ); vertical asymptote: x = 4 d) x = log(y โ 4) + 2 โ 10 xโ^2 = y โ 4. y = 10xโ^2 + 4 e) Domain of f โ^1 : (โโ, +โ) Range of f โ^1 : (4, +โ) f)
o (^) x
y
(b) f (x) = โ 3 x+
Solution: a) The domain of f : (โโ, +โ) b)
o x
y
c) Range: (โโ, 0); Horizontal asymptote: y = 0 d) x = โ 3 y+1^ โ 3 y+1^ = โx โ y + 1 = log 3 (โx) y = log 3 (โx) โ 1 e) Domain of f โ^1 : (โโ, 0)
(e) 2โx^ = 1. 5
Solution: 2 โx^ = 1. 5 โ โx = log 2 1. 5 , x = โ log 2 1. 5
(f) ex+3^ = ฯx
Solution: ex+3^ = ฯx ln ex+3^ = ln ฯx (x + 3) ln e = x ln ฯ x + 3 = x ln ฯ x(ln ฯ โ 1) = 3 x = (^) ln ฯ^3 โ 1
(g) 3 ยท 4 x^ + 4 ยท 2 x^ + 8 = 0
Solution: 3 ร 4 x^ + 4 ร 2 x^ + 8 = 0 Let y = 2x, then 3y^2 + 4y + 8 = 0, no solution.
- What rate of interest compounded annually is required to triple an investment in 10 years?
Solution: Suppose initial value = p, rate of interest compounded annually = r. p(1 + r)^10 = 3p (1 + r)^10 = 3 1 + r = 3 101 r = 3 101 โ 1
- The population of a southern city follows the exponential law: N (t) = N 0 ekt. If the population doubled in size over an 18-month period and the current population is 10,000, what will the population be 2 years from now?
Solution: Suppose t is in year. 2 N 0 = N 0 ekร^1.^5 2 = e^1.^5 k
- 5 k = ln 2 k = ln 2 1. 5 In two years, N (2) = 10000e^2 ร^ ln 2^1.^5 = 10000eln 2^
(^43) = 10000 ร 2 43 โ 25200
