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 x+log 4

Verified step by step guidance

1. Step 1: Use the properties of logarithms to combine the right side of the equation. The property $\log a+\log b=\log (ab)$ allows us to combine the logs: $\log x+\log 4=\log (4x)$.
2. Step 2: Set the equation $\log (x+4)=\log (4x)$. Since the logarithms are equal, their arguments must be equal: $x+4=4x$.
3. Step 3: Solve the equation $x+4=4x$ for $x$. Subtract $x$ from both sides to get $4=3x$.
4. Step 4: Divide both sides by 3 to isolate $x$: $x=\frac{4}{3}$.
5. Step 5: Check the solution to ensure it is in the domain of the original logarithmic expressions. Since $x=\frac{4}{3}$ is positive, it is valid for the logarithmic expressions $\log (x+4)$ and $\log x$.

Key Concepts

Logarithmic Properties

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.

Domain of Logarithmic Functions

The domain of a logarithmic function is restricted to positive real numbers. This means that any argument of a logarithm must be greater than zero. When solving logarithmic equations, it is crucial to check the solutions against the original equation to ensure they fall within this domain, as extraneous solutions may arise during the solving process.

Decimal Approximation

In many cases, logarithmic equations yield exact solutions that may not be easily interpretable. Therefore, using a calculator to find a decimal approximation can provide a more practical understanding of the solution. This involves evaluating the logarithmic expression numerically and rounding the result to a specified number of decimal places, often two for clarity.