Solve each equation. Give solutions in exact form. See Examples 5–9. log(x + 25) = log(x + 10) + log 4

Verified step by step guidance

1. Start by using the properties of logarithms to combine the logs on the right side of the equation. Recall that \( \log a + \log b = \log(ab) \). Therefore, rewrite the equation as \( \log(x + 25) = \log((x + 10) \cdot 4) \).
2. Simplify the right side of the equation: \( \log((x + 10) \cdot 4) = \log(4x + 40) \). Now the equation is \( \log(x + 25) = \log(4x + 40) \).
3. Since the logarithms are equal, their arguments must be equal as well. Set the arguments equal to each other: \( x + 25 = 4x + 40 \).
4. Solve the equation \( x + 25 = 4x + 40 \) for \( x \). Start by isolating \( x \) on one side. Subtract \( x \) from both sides to get \( 25 = 3x + 40 \).
5. Continue solving for \( x \) by subtracting 40 from both sides to get \( 25 - 40 = 3x \). Then, divide both sides by 3 to solve for \( x \).

Key Concepts

Properties of Logarithms

Understanding the properties of logarithms is essential for solving logarithmic equations. Key properties include the product rule, which states that log(a) + log(b) = log(ab), and the quotient rule, which states that log(a) - log(b) = log(a/b). These properties allow us to combine or separate logarithmic expressions, facilitating the solution process.

Isolating the Variable

Isolating the variable is a fundamental algebraic technique used to solve equations. In the context of logarithmic equations, this often involves manipulating the equation to express the variable in terms of constants. This step is crucial for finding the exact solutions, as it allows us to eliminate logarithmic terms and simplify the equation.

Exponential Form

Converting logarithmic equations to exponential form is a key step in solving them. The equation log_b(a) = c can be rewritten as a = b^c. This transformation is vital for finding the values of the variable, as it allows us to work with simpler algebraic expressions and directly solve for the unknown.