Solving a Logarithmic Equation Step-by-Step with Explanation

Question:

$$ \log x = \log \left( \frac{(20.48)^2 \times 2.412}{754.3} \right) $$

Answer:

$$ x = \frac{(20.48)^2 \times 2.412}{754.3} $$

Explain:

To solve for x, evaluate the expression inside the logarithm and eliminate the logarithm using the property of equality.

1. Start with the equation:
   $$ \log x = \log \left( \frac{(20.48)^2 \times 2.412}{754.3} \right) $$
2. Apply the property \log a = \log b \Rightarrow a = b:
   $$ x = \frac{(20.48)^2 \times 2.412}{754.3} $$
3. Calculate (20.48)^2:
   $$ (20.48)^2 = 419.4304 $$
4. Multiply by 2.412:
   $$ 419.4304 \times 2.412 = 1011.8856048 $$
5. Divide by 754.3:
   $$ x = \frac{1011.8856048}{754.3} \approx 1.341 $$

Summary:

Substitute the values into the expression and perform the arithmetic. The logarithm is removed by equating the arguments.

Key Concepts:

• Logarithmic Equations
• Equality Property of Logarithms: \log a = \log b \Rightarrow a = b
• Basic Arithmetic Operations

Related Formulas:

• Equality Rule: \log a = \log b \Rightarrow a = b
• Exponentiation Rule: \log x = y \Rightarrow x = 10^y

Application Scenarios:

1. Growth rate calculations in finance
2. Exponential decay problems
3. pH level computations in chemistry

Common Mistakes:

• Incorrect calculation of powers
• Misuse of logarithmic properties
• Errors in simplifying arithmetic expressions