Consider the function f(t)=logt(e), for t>0, t≠1.
(a) Use the change of base formula to find a constant a such that
f(t)=a/ln(t)
a=
(b) Evaluate f(e^2)=
.
(c) For what value(s) of t is f(t)=2?
.
log_t(e) = ln(e)/ln(t) = 1/ln(t)
a = 1
f(e^2) = 1/ln(e^2) = 1/2
f(t) = 2
1/ln(t) = 2
ln(t) = 1/2
t = √e
(a) To find the constant a, we can use the change of base formula for logarithms. The change of base formula states that log base a of b is equal to log base c of b divided by log base c of a. In this case, we are given f(t) = logt(e), and we want to express it as a/ln(t).
Using the change of base formula, we can rewrite f(t) as:
f(t) = (ln(e)) / (ln(t))
Since ln(e) is equal to 1, we can simplify the expression to:
f(t) = 1 / (ln(t))
Comparing this with a/ln(t), we can see that a = 1.
(a) a = 1
(b) To evaluate f(e^2), we substitute e^2 into the function f(t):
f(e^2) = 1 / (ln(e^2))
Using the property of logarithms that log base a of b^c is equal to c times log base a of b, we can rewrite the expression as:
f(e^2) = 1 / (2 * ln(e))
Since ln(e) is equal to 1, we have:
f(e^2) = 1 / (2 * 1)
Simplifying the expression gives:
f(e^2) = 1/2
(b) f(e^2) = 1/2
(c) To find the values of t for which f(t) is equal to 2, we can set the expression 1/(ln(t)) equal to 2 and solve for t.
1/(ln(t)) = 2
Multiply both sides by ln(t):
1 = 2 * ln(t)
Divide both sides by 2:
ln(t) = 1/2
Exponentiate both sides using the property that e^(ln(x)) = x:
t = e^(1/2)
The value of t can be written as the square root of e:
t = sqrt(e)
(c) t = sqrt(e)