Solve each equation. Give solutions in exact form. log(x + 25) = log(x + 10) + log 4

Verified step by step guidance

1. Recall the logarithm property that allows you to combine the sum of logarithms: \(\log a + \log b = \log (a \times b)\). Apply this to the right side of the equation \(\log(x + 10) + \log 4\) to rewrite it as a single logarithm.
2. Rewrite the equation using the property: \(\log(x + 25) = \log \big((x + 10) \times 4\big)\).
3. Since the logarithms on both sides have the same base (common logarithm, base 10), set their arguments equal to each other: \(x + 25 = 4(x + 10)\).
4. Solve the resulting linear equation for \(x\): first expand the right side to get \(x + 25 = 4x + 40\), then isolate \(x\) by moving terms to one side.
5. Check the solution(s) by substituting back into the original logarithmic expressions to ensure the arguments of the logarithms are positive, since the logarithm of a non-positive number is undefined.

Key Concepts

Properties of Logarithms

Understanding the properties of logarithms, such as the product, quotient, and power rules, is essential. In this problem, the product rule (log a + log b = log(ab)) allows combining the right side into a single logarithm, simplifying the equation for easier solving.

Solving Logarithmic Equations

Solving logarithmic equations involves rewriting the equation so that the logs on both sides have the same base, then equating their arguments. This step transforms the logarithmic equation into an algebraic one, which can be solved using standard algebraic methods.

Domain Restrictions of Logarithmic Functions

Logarithmic functions are only defined for positive arguments. When solving equations involving logs, it is crucial to check that the solutions make the arguments inside the logarithms positive, ensuring the solutions are valid within the domain.