Solve each logarithmic equation in Exercises 49–92. Be sure to reject any value of x that is not in the domain of the original logarithmic expressions. Give the exact answer. Then, where necessary, use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. 2 log x−log 7=log 112

Verified step by step guidance

1. Start with the given equation: \$2 \log x - \log 7 = \log 112$.
2. Use the logarithm power rule to rewrite \$2 \log x\( as \)\log x^2\(, so the equation becomes \)\log x^2 - \log 7 = \log 112$.
3. Apply the logarithm subtraction rule: \(\log a - \log b = \log \left( \frac{a}{b} \right)\), to combine the left side into a single logarithm: \(\log \left( \frac{x^2}{7} \right) = \log 112\).
4. Since \(\log A = \log B\) implies \(A = B\) (assuming the same base and valid domains), set the arguments equal: \(\frac{x^2}{7} = 112\).
5. Solve for \(x^2\) by multiplying both sides by 7, then take the square root of both sides to find \(x\). Remember to check the domain restrictions for logarithms: \(x\) must be positive.

Key Concepts

Properties of Logarithms

Understanding the properties of logarithms, such as the product, quotient, and power rules, is essential for simplifying and combining logarithmic expressions. For example, the difference of logs can be rewritten as the log of a quotient: log a - log b = log(a/b). These properties allow the equation to be manipulated into a solvable form.

Domain of Logarithmic Functions

The domain of a logarithmic function includes only positive real numbers because the logarithm of zero or a negative number is undefined. When solving logarithmic equations, it is crucial to check that the solutions fall within the domain of the original expressions to avoid extraneous or invalid answers.

Solving Logarithmic Equations

Solving logarithmic equations often involves rewriting the equation using logarithm properties, isolating the logarithmic expression, and then converting the logarithmic form to its equivalent exponential form. This process helps find exact solutions, which can then be approximated using a calculator if needed.