Convert Q = 2e^8t to form Q = ab^t
Round all calculated values to three decimal places
Q=
my answer I got 2(2.980.9579)^t
I don't think this is correct.
I took 2e^8t and tried solving for just e^8 then tried to convert... It turned out messy
a b^t = 2 e^(8t)
ln a + t ln b = ln 2 + 8 t
a = 2
and
ln b = 8
so b = e^8 = 2980
so I agree
To convert the expression Q = 2e^8t to the form Q = ab^t, let's break down the process step by step.
Step 1: Rewrite the expression Q = 2e^8t.
In this equation, the constant term is 2, and the exponential term is e^8t.
Step 2: Determine the value of e^8t.
To solve for e^8t, we need to isolate it by dividing both sides of the equation by 2:
Q/2 = e^8t
Step 3: Convert e^8t to the form ab^t.
To rewrite e^8t as ab^t, we need to find a base, b, such that e^8t = b^t.
Step 4: Determine the value of the base, b.
In mathematics, there is a fundamental relationship between the natural logarithm (ln) and the exponential function. It states that: e^x = y is equivalent to ln(y) = x.
Applying this relationship, we can take the natural logarithm of both sides of the equation from Step 3:
ln(Q/2) = ln(e^8t)
Step 5: Simplify the right side of the equation.
Using the relationship mentioned in Step 4, we can simplify ln(e^8t) to just 8t:
ln(Q/2) = 8t
Step 6: Rewrite the equation using the properties of logarithms.
To eliminate the natural logarithm, we can exponentiate both sides of the equation with a base of e:
e^(ln(Q/2)) = e^(8t)
Step 7: Simplify the left side of the equation.
The exponential of the natural logarithm is simply the argument itself:
Q/2 = e^(8t)
Step 8: Rewrite the equation in the desired form.
Since we want Q = ab^t, we can rewrite the equation as:
Q = 2 * e^(8t/2)
Step 9: Round all calculated values to three decimal places.
To adhere to the rounding requirement, we need to evaluate e^(8t/2) and round the result to three decimal places.
Therefore, the final conversion of Q = 2e^8t to the desired form Q = ab^t is:
Q = 2 * e^(8t/2), where the calculated value of e^(8t/2) should be rounded to three decimal places.