It will take about 22 years. It will occur in 2022.
Practice makes perfect
a
Let's analyze the given equation.
n=35[log_4(t+2)]
The variable n is the number of pet owners in thousands after t years after 2000. We are asked to find the number of pet owners in 2010. Therefore, we substitute 10 for t into the given equation.
Now, we will use the Change of Base Formula.
y=log_a b ⇒ y=log_()darkorangec b/log_()darkorangec a
This formula allows us to change the base of a logarithm. In our case, we will change the base to 10 to calculate the value more easily using a calculator.
n=35[log_4(12)] ⇒ n=35*log_(10)12/log_(10)4
Finally, using a calculator, we can get that n≈ 62.737. Therefore, in 2010 there was around 62.737 thousands of pet owners, or simply just 62 737 pet owners.
b
Since we want to know the year in which there will be 80 thousands pet owners, we will substitute 80 for n into the given equation.
To solve the equation for E, we will use the definition of a logarithm. Notice also that log x is equal to log_(10)x for all x.
y=log_b x ⇔ x= b^y
This definition tells us how to rewrite the logarithm equivalent of y as an exponential equation. The argument x is equal to b raised to the power of y. Let's do it!
2.29=log_4( t+2) ⇔ t+2= 4^(2.29)
Next, we will solve it for t.
