Section 7.4: Applications of Logs

This section covers real-world applications of logarithms, including population growth, pH calculations, and radioactive decay.

Example 1: Population Growth

The population of a town grows according to \( P(t) = 500 \cdot 2^{t/5} \). Find the time t when population reaches 2000.

Step 1: Set \( 2000 = 500 \cdot 2^{t/5} \)

Step 2: Divide both sides: \( 4 = 2^{t/5} \)

Step 3: Apply log base 2: \( \log_2 4 = t/5 \)

Step 4: Evaluate: \( 2 = t/5 \) → \( t = 10 \) years

A solution has a hydrogen ion concentration of \( 3.2 \times 10^{-5} \) M. Find its pH.

Step 1: pH = \( -\log[H^+] = -\log(3.2 \times 10^{-5}) \)

Step 2: Split log: \( -(\log 3.2 + \log 10^{-5}) = -(0.505 - 5) = 4.495 \)

Step 3: Rounded: pH ≈ 4.50

The half-life of a substance is 8 hours. If initial mass is 100 g, find remaining mass after 20 hours.

Step 1: Use formula: \( m(t) = m_0 (1/2)^{t/8} = 100 \cdot (1/2)^{20/8} \)

Step 2: Simplify exponent: \( 20/8 = 2.5 \)

Step 3: Compute: \( m = 100 \cdot (1/2)^{2.5} = 100 \cdot 0.1768 ≈ 17.68 \) g

A bacteria culture doubles every 3 hours. If initial count is 200, when will it reach 1600?
Find the pH of a solution with \( [H^+] = 4.5 \times 10^{-6} \) M
A radioactive isotope has a half-life of 12 hours. If starting mass is 50 g, how much remains after 30 hours?
A population grows according to \( P(t) = 1000 \cdot 3^{t/4} \). Find t when P = 27000
Solve for x: \( 5^x = 125 \)