Write each equation in its equivalent exponential form. Then solve for x. log₃ (x-1) = 2

Verified step by step guidance

1. Recall the definition of logarithm: if \( \log_b a = c \), then the equivalent exponential form is \( b^c = a \).
2. Apply this definition to the given equation \( \log_3 (x-1) = 2 \). Rewrite it as \( 3^2 = x - 1 \).
3. Calculate the exponential expression \( 3^2 \) (you can leave it as \( 3^2 \) for now if you prefer).
4. Set up the equation \( 3^2 = x - 1 \) and solve for \( x \) by adding 1 to both sides: \( x = 3^2 + 1 \).
5. Check the solution by substituting \( x \) back into the original logarithmic equation to ensure the argument \( x - 1 \) is positive and the equation holds true.

Key Concepts

Definition of Logarithms

A logarithm answers the question: to what exponent must the base be raised to produce a given number? For example, log_b(a) = c means b^c = a. Understanding this definition is essential to rewrite logarithmic equations in exponential form.

Converting Logarithmic Equations to Exponential Form

To solve logarithmic equations, rewrite them in exponential form using the equivalence log_b(a) = c ⇔ b^c = a. This conversion simplifies solving for the variable inside the logarithm by turning the equation into an exponential equation.

Solving Exponential Equations

Once the equation is in exponential form, solve for the variable by isolating it. This may involve basic algebraic operations such as addition, subtraction, multiplication, division, or taking roots, depending on the equation's complexity.