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. 2 log x=log 25

Key Concepts

Logarithmic Functions

Logarithmic functions are the inverses of exponential functions and are defined as the power to which a base must be raised to obtain a given number. The logarithm of a number is only defined for positive values, which is crucial when solving logarithmic equations. Understanding the properties of logarithms, such as the product, quotient, and power rules, is essential for manipulating and solving these equations.

Domain of Logarithmic Functions

The domain of a logarithmic function is restricted to positive real numbers. This means that any solution derived from a logarithmic equation must be checked to ensure it falls within this domain. If a solution yields a logarithm of a non-positive number, it must be rejected as invalid, which is a critical step in solving logarithmic equations.

Solving Logarithmic Equations

To solve logarithmic equations, one typically uses properties of logarithms to combine or simplify the expressions. In the given equation, 2 log x = log 25, one can apply the power rule to rewrite it as log x^2 = log 25, allowing for the comparison of arguments. After finding potential solutions, it is important to verify them against the domain restrictions to ensure they are valid.