Find all the real-number roots of each equation. In each...

Find all the real-number roots of each equation. In each case, give an exact expression for the root and also (where appropriate) a calculator approximation rounded to three decimal places.
$$\log _{10}(x+3)-\log _{10}(x-2)=2$$

Key Concepts

Logarithmic Functions

Logarithmic functions are the inverses of exponential functions. They are used to solve equations in which the unknown appears as the exponent, and they have properties that facilitate manipulation and simplification of expressions. Logarithms with a specific base represent the power to which that base must be raised to produce a given number.

Properties of Logarithms

Key properties of logarithms, such as the product rule, quotient rule, and power rule, allow us to combine or separate logarithmic expressions. In particular, the quotient rule, which states that log_b(A) - log_b(B) equals log_b(A/B), is useful for consolidating differences of logs into a single expression, thereby simplifying the process of solving logarithmic equations.

Domain of Logarithmic Functions

Any function containing a logarithm is defined only for positive arguments. This necessitates checking the domain of the logarithmic expressions to ensure that the values under each logarithm are positive. In solving logarithmic equations, it is important to recognize and apply these domain restrictions to rule out extraneous solutions.

Solving Logarithmic Equations

Solving logarithmic equations typically involves using the properties of logarithms to combine separate logarithmic terms into one, after which the equality can be rewritten in its exponential form. This process transforms the logarithmic equation into an algebraic equation, which can then be solved for the unknown variable. Finally, any solutions must be verified to ensure they comply with the domain restrictions of the original logarithmic expressions.

Numerical Approximation

While problems in logarithms often yield exact algebraic expressions for solutions, it is sometimes useful to express these in decimal form for practical applications. Numerical approximation involves using a calculator or computational method to estimate the value of the expressions to a given level of precision, while still providing the exact symbolic result alongside for clarity.

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