Question

1. interpret logarithmic functions in terms of time that it takes for a population with certain characteristics to rea

week 13 - (section 4.2)

• use logarithms to express the time that it takes for 1 unit that triples each year to increase to 70 units. solutio

Explanation:

Step1: Define the growth model

The population grows exponentially. The formula for exponential growth is \( P(t) = P_0 \cdot r^t \), where \( P_0 \) is the initial population, \( r \) is the growth rate, and \( t \) is time. Here, \( P_0 = 1 \), \( r = 3 \) (triples each year), and \( P(t)=70 \). So the equation is \( 70 = 1 \cdot 3^t \).

Step2: Apply logarithm to solve for t

Take the logarithm of both sides. Using the natural logarithm (ln) or common logarithm (log) works. Let's use natural logarithm. Take \( \ln \) of both sides: \( \ln(70)=\ln(3^t) \). By the logarithm power rule \( \ln(a^b)=b\ln(a) \), this becomes \( \ln(70)=t\ln(3) \). Then solve for \( t \): \( t = \frac{\ln(70)}{\ln(3)} \) (or using common logarithm \( t=\frac{\log(70)}{\log(3)} \), since \( \frac{\ln(a)}{\ln(b)}=\frac{\log(a)}{\log(b)} \) by change - of - base formula).

Answer:

The time \( t \) is given by \( t=\frac{\ln(70)}{\ln(3)} \) (or \( t = \log_3(70) \) using the definition of logarithm \( \log_b(a)=\frac{\ln(a)}{\ln(b)} \))