Exponential Logarithmic and Other Common Functions

Exponential Logarithmic and Other Common Functions
Order 5889231
Exponential Logarithmic and Other Common Functions
Exponential Functions
For questions 273 to 275, find the function value for f(x) = b
x (b ≠ 1, b > 0).
Refer to the following rules for exponents, as needed.
x
1 = x
x
0 = 1, x ≠ 0
0
0
is undefined
(x
n)
p = x
np (power of a power)
(power of a quotient)
(xy)
p = x
p y
p (power of a product)
x
mx
n = x
m n (product rule)
(quotient rule),
provided, in all cases, that division by zero does not occur; and when restricted
to real numbers, that even roots of negative quantities do not occur.
273. f(x) = 5
x
; f(4)
274.
275.
For questions 276 and 277, (a) state the domain and range, (b) find the zeros, (c)
determine the asymptotes, (d) find the intercepts, (e) discuss increasing and
decreasing behavior, and (f) discuss behavior as x approaches ±∞. Refer to the
following guidelines, as needed.
The graph of f(x) = b
x (b ≠ 1,b > 0) is a smooth, continuous curve. The graph
passes through the points (0,1) and (1,b) and is located in the first and second
quadrants only. The domain is R, and the range is (0,∞). The y-intercept is 1. It
has no x-intercepts. The x-axis is a horizontal asymptote.
Furthermore, the following hold:
• If b > 1, the function is increasing. As x approaches ∞, f(x) = b
x approaches
∞. As x approaches –∞, f(x) = b
x approaches 0, but never reaches 0.
• If 0 < b 0),
f(x) = b
x > 0, for all real numbers
f(0) = b
0 = 1
f(1) = b
1 = b
f(u) · f(v) = b
u · b
v = b
u v = f(u v)
(f(x))
p = (b
x)
p = b
xp = f(xp)
One-to-one property: f(u) = f(v) if and only if u = v; that is, b
u = b
v
if and only if
u = v.
278. Suppose .
279. Suppose f(x) = 2
x
. Find f(3) · f(2).
280. Suppose .
281. Suppose f(x) = e
x
. If e
u = e
21
, then u = ____________.
Logarithmic Functions
For questions 282 to 285, find the function value for f(x) = logb x, (b ≠ 1,b > 0).
282. f(x) = log5 x; f(625)
283. f(x) = log27 x; f(81)
Hint: Consider fractional exponents.
284.
285.
286. State the function g that defines the inverse of f.
(A) f(x) = log6 x
(B) f(x) = (1.035)
x
(C)
(D) f(x) = ln x
(E) f(x) = log x
For questions 287 and 288, (a) state the domain and range, (b) find the zeros, (c)
determine the asymptotes, (d) find the intercepts, (e) discuss increasing and
decreasing behavior, and (f) discuss behavior as x approaches 0 or ∞. Refer to
the following guidelines, as needed.
The graph of f(x) = logb x (b ≠ 1, b > 0) is a smooth, continuous curve. The graph
passes through (1,0) and (b, 1) and is located in the first and fourth quadrants
only. The domain is (0,∞), and the range is R. The x-intercept is 1. It has no yintercepts. The y-axis is a vertical asymptote.
Furthermore, the following hold:
• If b > 1, the function is increasing. As x approaches ∞, f(x) = logb x
approaches ∞. As x approaches 0, f(x) = logb
, x approaches –∞.
• If 0 < b < 1, the function is decreasing. As x approaches 0, f(x) = b
x
approaches ∞. As x approaches ∞, f(x) = logb x approaches –∞.
287. f(x) = log6 x
288.
In questions 289 to 292, evaluate using the following properties of logarithmic
functions, as needed.
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