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UNIT FIVE
UNIVERSITY OF THE PEOPLE
1201-01: COLLEGE ALGEBRA
, 2
Task 1
(i) Exponential and logarithmic functions are closely related. An exponential function is of
the form f(x) = a * b^x, where a is a constant (often the initial value), b is the positive base
(b > 0, b ≠ 1), and x is the exponent. The variable x can be any real number. The domain is
all real numbers, and the range is (0, ∞) if a > 0. A logarithmic function is the inverse of an
exponential function: if y = b^x, then x = log_b(y). So log_b(x) = y means b^y = x. Here x
must be positive, so the domain of a logarithmic function is (0, ∞) and the range is all real
numbers. The base b is the same positive number not equal to 1. They are inverses:
applying one after the other returns the original input.
(ii) The differences:
· Exponential function: variable in the exponent, e.g., f(x) = 2^x. It grows (or decays) at a
rate proportional to its current value. Graph: passes through (0,1), has a horizontal
asymptote at y = 0 as x → -∞, and increases rapidly for x > 0.
· Logarithmic function: variable inside the log, e.g., f(x) = log_2(x). It is the inverse of
exponential. Graph: passes through (1,0), has a vertical asymptote at x = 0, and increases
very slowly for large x.
· Power function: variable raised to a constant power, e.g., f(x) = x^2. It grows at a
polynomial rate. Graph: passes through (0,0), symmetric about y-axis, no asymptotes.
Growth patterns: Exponential eventually outpaces any power function; logarithmic grows
slower than any positive power function. Special points: exponential has y-intercept (0,a);
logarithmic has x-intercept (1,0); power functions often have intercept at origin.
(iii) A function exhibits exponential growth if it can be written as f(x) = a * b^x with b > 1,
and its rate of change is proportional to its current value. In practical terms, the quantity
increases by a fixed percentage over equal time intervals.
(iv) Exponential functions grow much faster than logarithmic functions. As x increases, an
exponential function like 2^x increases without bound rapidly, while a logarithmic function
like log_2(x) increases very slowly (its derivative decreases). For large x, exponential
dominates.
(v) Observations from graphs:
● Intersection Points: The red and green curves intersect twice. For a brief window,
the quadratic growth (x^2) is faster, but the exponential growth (e^x) quickly
dominates after the second intersection.
UNIT FIVE
UNIVERSITY OF THE PEOPLE
1201-01: COLLEGE ALGEBRA
, 2
Task 1
(i) Exponential and logarithmic functions are closely related. An exponential function is of
the form f(x) = a * b^x, where a is a constant (often the initial value), b is the positive base
(b > 0, b ≠ 1), and x is the exponent. The variable x can be any real number. The domain is
all real numbers, and the range is (0, ∞) if a > 0. A logarithmic function is the inverse of an
exponential function: if y = b^x, then x = log_b(y). So log_b(x) = y means b^y = x. Here x
must be positive, so the domain of a logarithmic function is (0, ∞) and the range is all real
numbers. The base b is the same positive number not equal to 1. They are inverses:
applying one after the other returns the original input.
(ii) The differences:
· Exponential function: variable in the exponent, e.g., f(x) = 2^x. It grows (or decays) at a
rate proportional to its current value. Graph: passes through (0,1), has a horizontal
asymptote at y = 0 as x → -∞, and increases rapidly for x > 0.
· Logarithmic function: variable inside the log, e.g., f(x) = log_2(x). It is the inverse of
exponential. Graph: passes through (1,0), has a vertical asymptote at x = 0, and increases
very slowly for large x.
· Power function: variable raised to a constant power, e.g., f(x) = x^2. It grows at a
polynomial rate. Graph: passes through (0,0), symmetric about y-axis, no asymptotes.
Growth patterns: Exponential eventually outpaces any power function; logarithmic grows
slower than any positive power function. Special points: exponential has y-intercept (0,a);
logarithmic has x-intercept (1,0); power functions often have intercept at origin.
(iii) A function exhibits exponential growth if it can be written as f(x) = a * b^x with b > 1,
and its rate of change is proportional to its current value. In practical terms, the quantity
increases by a fixed percentage over equal time intervals.
(iv) Exponential functions grow much faster than logarithmic functions. As x increases, an
exponential function like 2^x increases without bound rapidly, while a logarithmic function
like log_2(x) increases very slowly (its derivative decreases). For large x, exponential
dominates.
(v) Observations from graphs:
● Intersection Points: The red and green curves intersect twice. For a brief window,
the quadratic growth (x^2) is faster, but the exponential growth (e^x) quickly
dominates after the second intersection.
