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 logarithmic equation: $2logx-log7=log112$.
2. Use the logarithmic property that allows you to rewrite coefficients as exponents: $2logx=log{x}^2$. So the equation becomes $log{x}^2-log7=log112$.
3. Apply the logarithmic subtraction rule: $loga\; -logb\; =log\backslash left(\backslash frac\{a\}\{b\}\backslash right)$. This transforms the left side into $log\backslash left(\backslash frac\{x^2\}\{7\}\backslash right)$, so the equation is now $log\backslash left(\backslash frac\{x^2\}\{7\}\backslash right)\; =log112$.
4. Since the logarithms on both sides have the same base and are equal, set their arguments equal: $\backslash frac\{x^2\}\{7\}\; =\; 112$.
5. Solve for $x$ by multiplying both sides by 7 to get $x^2\; =\; 784$, then take the square root of both sides to find $x\; =\; \backslash pm\; \backslash sqrt\{784\}$. Remember to check which solutions are valid by ensuring the arguments of the original logarithms are 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). This helps in transforming the given equation 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 to reject extraneous or invalid answers.

Solving Logarithmic Equations

Solving logarithmic equations often involves rewriting the equation using logarithm properties, then converting it to an exponential form to isolate the variable. After finding potential solutions, verify them against the domain restrictions and, if needed, use a calculator for decimal approximations.