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

Key Concepts

Logarithmic Properties

Understanding the properties of logarithms is essential for solving logarithmic equations. Key properties include the product rule (log(a) + log(b) = log(ab)), the quotient rule (log(a) - log(b) = log(a/b)), and the power rule (n * log(a) = log(a^n)). These properties allow us to combine or simplify logarithmic expressions, making it easier to isolate the variable.

Domain of Logarithmic Functions

The domain of a logarithmic function is restricted to positive real numbers. This means that the arguments of the logarithmic expressions must be greater than zero. When solving logarithmic equations, it is crucial to check the solutions against the original expressions to ensure they fall within this domain, as any extraneous solutions can lead to invalid results.

Decimal Approximation

After finding the exact solution to a logarithmic equation, it may be necessary to provide a decimal approximation. This involves using a calculator to evaluate the logarithmic expressions or the resulting values to a specified number of decimal places. In this case, rounding to two decimal places ensures that the solution is presented in a clear and concise manner, suitable for practical applications.