Exponential Logarithmic and Other Common Functions

Exponential Logarithmic and Other Common Functions

. f (4) = 5

4 = 625

274. f = = 3

4 = 81

275. f(–6) = = 2

6 = 64

276. (a) The domain is R and the range is (0,∞); (b) There are no zeros; (c) x =

0 is a horizontal asymptote; (d) The y-intercept is y = 1 and there are no xintercepts; (e) b = 3 > 1, therefore f is increasing on R; (f) As x approaches ∞, f

(x) approaches ∞ and as x approaches –∞, f (x) approaches 0.

277. (a) The domain is R and the range is (0,∞); (b) There are no zeros; (c) x =

0 is a horizontal asymptote; (d) The y-intercept is 1 and there are no x-intercepts;

(e) b = 0). Thus, g(x) = 6

x

is the inverse of f (x) = log6 x.

Exponential Logarithmic and Other Common Functions

(B) The logarithmic function f (x) = logb x is the inverse of the

exponential function g(x) = b

x

(b ≠ 1,b > 0). Thus, g(x) = log1.035 x is

the inverse of f (x) = (1.035)

x

.

(C) g(x) = x is the inverse of f (x) =

(D) f (x) = ln x is the natural logarithmic function that has base e; that is f

(x) = ln x = loge x, where e is the irrational constant approximately

equal to 2.718281828. Thus, its inverse is g(x) = e

x

. See the

following figure.

(E) f (x) = log x is the common logarithmic function that has base 10;

that is f (x) = log x = log10 x. Thus, its inverse is g(x) = 10

x

.

287. (a) The domain is (0,∞) and the range is (–∞,∞); (b) x = 1 is the only zero;

(c) The y-axis is a vertical asymptote; (d) The x-intercept is (1,0) and there are

no y-intercepts; (e) b = 6 > 1, so f (x) is increasing on (0,∞); (f) b = 6 > 1, so as x

approaches 0, f (x) approaches –∞ and as x approaches ∞, f (x) approaches ∞.

288. (a) The domain is (0,∞) and the range is (–∞,∞); (b) x = 1 is the only zero;

(c) The y-axis is a vertical asymptote; (d) The x-intercept is (1,0) and there are

no y-intercepts; (e) b = < 1, so f (x) is decreasing on (0,∞); (f) b = < 1, so as

x approaches 0, f (x) approaches ∞ and as x approaches ∞, f (x) approaches –∞.

289. f(e

10) = lne

10 = 10lne = 10–1 = 10

290. g(64

20) = log2 64

20 = 20log2 64 = 20(6) = 120

291. = log(100) – log(0.00000l) = log(10

2) – log(10

–6)

= 2 – (–6) = 2 6 = 8

292. g(8 · 32) = log2 (8 · 32) = log2 8 log2 32 = 3 5 = 8

293. Using the one-to-one property, u = 450.

294. (A) log8(32,768) = = 5

(B) (0.0016) = = 4

(C) log2(4,096) = = 12

(D) log1.05(2.5) = ≈ 18.78

(E) log2(400) ≈ 8.64

295.

296.

297.

298.

299.

300. Substituting A0 = 20 and k = 5,730 into the formula, A(t) = Ao

, and

evaluating at t = 5,000, omitting units for convenience, yields A(5,000) = 20

≈ 10.92 grams.

301. Evaluating the formula for x = 7.6 × 10

–4 yields pH of diet soda Z = f(7.6 ×

10

–4) = –log10(7.6 × 10

–4) ≈ 3.12.

302. To find the rate, compounded annually, that will double an investment of

$50,000 in 20 years, substitute P = $100,000, P = $50

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