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(3x−3)=log(x+1)+log 4

Verified step by step guidance

1. Start with the given equation: \(\log(3x - 3) = \log(x + 1) + \log 4\).
2. Use the logarithm property that states \(\log a + \log b = \log(ab)\) to combine the right side: \(\log(3x - 3) = \log(4(x + 1))\).
3. Since the logarithms on both sides have the same base and are equal, set their arguments equal: \$3x - 3 = 4(x + 1)$.
4. Solve the resulting linear equation for \(x\): expand the right side to get \$3x - 3 = 4x + 4\(, then isolate \)x$ by moving terms appropriately.
5. Check the solution(s) by substituting back into the original logarithmic expressions to ensure the arguments are positive, because the domain of \(\log\) requires the input to be greater than zero.

Key Concepts

Properties of Logarithms

Understanding the properties of logarithms, such as the product rule log(a) + log(b) = log(ab), is essential for combining or simplifying logarithmic expressions. This allows you to rewrite the equation in a more manageable form to isolate the variable.

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. Identifying and restricting the domain ensures that any solutions found are valid within the original equation.

Solving Logarithmic Equations

Solving logarithmic equations often involves rewriting the equation in exponential form or using properties of logarithms to isolate the variable. After simplification, you solve the resulting algebraic equation and verify solutions against the domain restrictions.