91Ó°ÊÓ

The quotient rule for logarithms is a key property that helps simplify the logarithms of fractions. It states that the logarithm of a quotient is the logarithm of the numerator minus the logarithm of the denominator. In mathematical terms, this is written as: \ \( \log _{10} \frac{a}{b}= \log_{10}a- \log_{10}b \). For the given problem, \( \log _{10} \frac{9}{4} \), we apply this property to rewrite it as: \ \( \log_{10} 9 - \log_{10} 4 \.\) This makes it easier to handle if we know or can find the logarithms of simpler numbers like 2 and 3. This rule simplifies complex logarithmic calculations. Try identifying other opportunities to use this in your homework.

Another crucial property is the power rule for logarithms, which allows us to handle logarithms of numbers raised to a power. It states: \ \( \log_{10} a^b = b \cdot \log_{10} a \). This means you can move the exponent in front as a multiplier. In the problem, we use this with: \ \( \log_{10} 3^2 - \log_{10} 2^2 = 2 \cdot \log_{10} 3 - 2 \cdot \log_{10} 2 \.\) Doing so takes the exponential complexity out of the logarithms, making it simpler to substitute known values. This rule is exceptionally handy when you encounter exponents in logarithmic expressions.

Substitution is a problem-solving technique where known values are substituted into an expression. In our example, we substitute the known logarithm values for 2 and 3 into the expression we derived. Since we know: \ \( \log_{10} 2 = 0.3010 \) and \ \( \log_{10} 3 = 0.4771 \,\) we replace these in the equation: \ \( 2 \cdot \log_{10} 3 - 2 \cdot \log_{10} 2 = 2 \cdot 0.4771 - 2 \cdot 0.3010. \) This step simplifies the solution process, making it purely arithmetic. Finally, we perform the calculations: \ \( 2 \cdot 0.4771 = 0.9542 \) and \ \( 2 \cdot 0.3010 = 0.6020 \), then find the difference: \ \( 0.9542 - 0.6020 = 0.3522 \.\) Substitution is a core method to make complex equations more manageable.