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?

Explanation:

Step1: Define the growth formula

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

Step2: Apply logarithm to solve for t

Take log (base 2) on both sides: $\log_2(8)=\log_2(2^t)$. Using $\log_a(a^x)=x$, we get $t = \log_2(8)$. Alternatively, using natural log: $\ln(8)=t\ln(2)$, so $t=\frac{\ln(8)}{\ln(2)}$.

Answer:

The time \( t \) (in years) can be expressed as \( t = \log_2(8) \) (or \( t=\frac{\ln(8)}{\ln(2)} \), which simplifies to \( t = 3 \)).