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. Start with the given equation: $\log (x+4)=\log \left(x\right)+\log \left(4\right)$.
2. Use the logarithmic property that $\log \left(a)\; +\log \right(b)\; =\log (ab)$ to combine the right side: $\log \left(x\right)+\log \left(4)\; =\log \right(4x)$.
3. Rewrite the equation as $\log \left(x+\; 4)\; =\log \right(4x)$.
4. Since the logarithms are equal and the log function is one-to-one, set the arguments equal: $x+\; 4\; =\; 4x$.
5. Solve the equation $x+\; 4\; =\; 4x$ for $x$, then check the solution(s) to ensure they are in the domain of the original logarithmic expressions (i.e., arguments inside the logs must be positive).

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 terms on one side to simplify the equation and solve for x.

Domain of Logarithmic Functions

The domain of a logarithmic function includes only positive arguments. When solving logarithmic equations, it is crucial to check that the solutions do not make any logarithm’s argument zero or negative, as these values are not valid.

Solving Logarithmic Equations

Solving logarithmic equations often involves rewriting the equation using logarithm properties, then converting to exponential form or equating arguments when logs have the same base. This process helps isolate the variable and find exact solutions.