Use laws of logarithms to solve problems

Example 1: Using the Product Rule

Evaluate $\log_2(8 \times 4)$

Solution

Using the product rule: $\log_2(8 \times 4) = \log_2 8 + \log_2 4$ $= \log_2 2^3 + \log_2 2^2$ $= 3\log_2 2 + 2\log_2 2$ $= 3(1) + 2(1) = 5$

Example 2: Using the Quotient Rule

Evaluate $\log_3\left(\frac{27}{9}\right)$

Solution

Using the quotient rule: $\log_3\left(\frac{27}{9}\right) = \log_3 27 - \log_3 9$ $= \log_3 3^3 - \log_3 3^2$ $= 3\log_3 3 - 2\log_3 3$ $= 3(1) - 2(1) = 1$

Example 3: Using the Power Rule

Evaluate $\log_3 9^2$

Solution

Using the power rule: $\log_3 9^2 = 2 \log_3 9$ $= 2 \log_3 3^2$ $= 2 \times 2 \log_3 3$ $= 4(1) = 4$

Example 4: Simplifying with Given Logarithm Values

Given that $\log 2 = 0.3010$, $\log 3 = 0.4771$, and $\log 5 = 0.6990$, evaluate $\log 30$.

Solution

Write 30 as a product: $30 = 3 \times 10$

$\log 30 = \log(3 \times 10) = \log 3 + \log 10$ $= \log 3 + 1$ $= 0.4771 + 1 = 1.4771$

Example 5: Using Change of Base Formula

Evaluate $\log_2 32$

Solution

Using change of base (converting to base 10): $\log_2 32 = \frac{\log_{10} 32}{\log_{10} 2}$ $= \frac{\log_{10} 2^5}{\log_{10} 2}$ $= \frac{5\log_{10} 2}{\log_{10} 2} = 5$

When solving problems using laws of logarithms:

1. Identify which law or combination of laws applies
2. Apply the law(s) to expand or simplify the expression
3. Substitute known values (given logarithm values)
4. Calculate the final answer

Example: Solving for Unknown

If $\log_a x = \log_a 4 + \log_a 5$, find $x$.

Solution

Using the product rule in reverse: $\log_a x = \log_a 4 + \log_a 5 = \log_a(4 \times 5) = \log_a 20$

Therefore, $x = 20$.

Logarithms are used in calculating interest rates and loan payments at banks and mobile banking services (such as M-Pesa). For example, if you save Tsh 500,000 in a fixed deposit account with an annual interest rate of 5%, logarithms help determine exactly how many years it will take for your money to double. This is essential for planning savings and investments in everyday financial decisions in Tanzania.