Question

if one unit doubles every year, after 3 years we will have 8 units?
how can we express the number of years it takes one unit to grow to 8 units using logarithms?
a. log₃(8)
b. ln(8)
c. ln(2)
d. log₈(2)
e. none

Explanation:

Step1: Define the growth model

The growth is exponential. The formula for exponential growth is $N = N_0 \times 2^t$, where $N_0 = 1$ (initial units), $N = 8$ (final units), and $t$ is the time in years. So we have $8 = 1\times2^t$, which simplifies to $2^t = 8$.

Step2: Convert to logarithmic form

Recall that if $a^b = c$, then $b=\log_a c$. From $2^t = 8$, we can rewrite it as $t=\log_2 8$. We can also use natural logarithms: take the natural log of both sides of $2^t = 8$: $\ln(2^t)=\ln(8)$. Using the logarithm power rule $\ln(a^b)=b\ln(a)$, we get $t\ln(2)=\ln(8)$, so $t = \frac{\ln(8)}{\ln(2)}$ (which is equal to $\log_2 8$ by the change - of - base formula). Now let's check the options:

• Option A: $\log_3(8)$ is not related to our base (which is 2), so A is wrong.
• Option B: $\ln(8)$ is just the numerator in our expression for $t$, not $t$ itself, so B is wrong.
• Option C: $\ln(2)$ is the denominator in our expression for $t$, not $t$ itself, so C is wrong.
• Option D: $\log_8(2)$ is the reciprocal of $\log_2(8)$ (by the change - of - base formula, $\log_a b=\frac{1}{\log_b a}$), so D is wrong.

Answer:

E. None