In Exercises 101–104, write each equation in its equivalent exponential form. Then solve for x. log3 (x-1) = 2

Key Concepts

Logarithmic Functions

A logarithmic function is the inverse of an exponential function. It answers the question: to what exponent must a base be raised to produce a given number? In the equation log_b(a) = c, b is the base, a is the result, and c is the exponent. Understanding this relationship is crucial for converting logarithmic equations into exponential form.

Exponential Form

Exponential form is a way to express equations where a constant base is raised to a power. For example, the logarithmic equation log_b(a) = c can be rewritten in exponential form as b^c = a. This transformation is essential for solving logarithmic equations, as it allows us to isolate the variable and find its value.

Solving for x

Solving for x involves finding the value of the variable that satisfies the equation. In the context of logarithmic and exponential equations, this often requires isolating x on one side of the equation. Techniques may include applying inverse operations, simplifying expressions, and sometimes using properties of logarithms or exponents to facilitate the solution process.