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. log(x+4)−log 2=log(5x+1)

Verified step by step guidance

1. Start with the given equation: \(\log(x+4) - \log 2 = \log(5x+1)\).
2. Use the logarithmic property that states \(\log a - \log b = \log \left( \frac{a}{b} \right)\) to combine the left side: \(\log \left( \frac{x+4}{2} \right) = \log(5x+1)\).
3. Since \(\log A = \log B\) implies \(A = B\) (assuming the bases are the same and the arguments are positive), set the arguments equal: \(\frac{x+4}{2} = 5x + 1\).
4. Solve the resulting equation for \(x\) by multiplying both sides by 2 to clear the denominator: \(x + 4 = 2(5x + 1)\), then simplify and isolate \(x\).
5. Check the domain restrictions: ensure that the arguments of all logarithms are positive, so \(x + 4 > 0\) and \(5x + 1 > 0\). Reject any solution that does not satisfy these conditions.

Key Concepts

Properties of Logarithms

Understanding the properties of logarithms, such as the product, quotient, and power rules, is essential. In this problem, the quotient rule allows combining or separating logarithmic expressions, e.g., log(a) - log(b) = log(a/b), which simplifies the equation for easier solving.

Domain of Logarithmic Functions

The domain of a logarithmic function includes only positive arguments because the logarithm of zero or a negative number is undefined. When solving logarithmic equations, it is crucial to check that solutions keep all log arguments positive to ensure valid answers.

Solving Logarithmic Equations

Solving logarithmic equations often involves rewriting the equation using logarithm properties, then converting to an exponential form to isolate the variable. After finding potential solutions, verify each against the domain restrictions to reject extraneous roots.