1) √5^(8n) = 125^(n + 4)
a. n = 12
b. n = 4
c. n = –4
d. n = –12
2) Part 1: Describe the graphs of the functions f(x) = 2^x – 3 and g(x) = –2^x – 3. (3 points)
Part 2: Compare and contrast the domain and range of f(x) and g(x). (3 points)
a. n = 12
b. n = 4
c. n = –4
d. n = –12
2) Part 1: Describe the graphs of the functions f(x) = 2^x – 3 and g(x) = –2^x – 3. (3 points)
Part 2: Compare and contrast the domain and range of f(x) and g(x). (3 points)
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Rewriting the equation in terms of radical expressions:
(5^(1/2))^8n = (5^3)^(n + 4)
Applying the properties of exponents (a^m)^n = a^(m*n) on both sides we get:
5^(8n/2) = 5^(3*(n+4))
5^(4n) = 5^(3n+12)
Applying logarithms on both sides we get:
log(5^(4n)) = log(5^(3n+12))
4n*log(5) = (3n+12)*log(5)
Cancelling out log(5) we are left with:
4n =3n+12
n = 12
So, the answer for 1) is "a".
Now, regarding part 1 of question 2), they' re both symmetric with respect to the line y = -3, because:
lim f(x) when x tends to minus infinity is -3, and so it is for g(x).
As for part 2, the domain of both functions is <-infinity,+infinity> and the ranges are:
For f(x): <-3,+infinity>
For g(x): <-infinity,-3>
(5^(1/2))^8n = (5^3)^(n + 4)
Applying the properties of exponents (a^m)^n = a^(m*n) on both sides we get:
5^(8n/2) = 5^(3*(n+4))
5^(4n) = 5^(3n+12)
Applying logarithms on both sides we get:
log(5^(4n)) = log(5^(3n+12))
4n*log(5) = (3n+12)*log(5)
Cancelling out log(5) we are left with:
4n =3n+12
n = 12
So, the answer for 1) is "a".
Now, regarding part 1 of question 2), they' re both symmetric with respect to the line y = -3, because:
lim f(x) when x tends to minus infinity is -3, and so it is for g(x).
As for part 2, the domain of both functions is <-infinity,+infinity> and the ranges are:
For f(x): <-3,+infinity>
For g(x): <-infinity,-3>