Question: 7 — Solving Exponential Equations using Logarithms
LEARNING OBJECTIVE: Solve an exponential equation by applying logarithmic properties.
The equation $y = 2(3^t)$ shows the number of infected people from an outbreak
of the Norovirus. The variable $y$ represents the number of infected people, and
the variable $t$ represents time in weeks.
In how many weeks will the number of infected people reach 1,458?
a.) 6 weeks
b.) 5 weeks
c.) 12 weeks
d.) 10 weeks
1. **State the problem:** We want to find the time $t$ in weeks when the number of infected people $y$ reaches 1458, given the equation $$y = 2(3^t).$$
2. **Write the equation with the known value:**
$$1458 = 2(3^t).$$
3. **Isolate the exponential term:**
$$\frac{1458}{2} = 3^t$$
$$729 = 3^t.$$
4. **Use logarithms to solve for $t$:**
Take the logarithm base 3 of both sides:
$$\log_3(729) = \log_3(3^t).$$
5. **Apply logarithm power rule:**
$$\log_3(729) = t \cdot \log_3(3).$$
Since $\log_3(3) = 1$, this simplifies to:
$$\log_3(729) = t.$$
6. **Evaluate $\log_3(729)$:**
Since $729 = 3^6$, we have:
$$t = 6.$$
7. **Answer:** The number of infected people will reach 1458 in **6 weeks**.
**Final answer:** a.) 6 weeks