Solve each equation. Give solutions in exact form. log₆ (2x + 4) = 2

Verified step by step guidance

1. Recall the definition of logarithm: if \(\log_b A = C\), then it is equivalent to the exponential form \(b^C = A\). Here, the base is 6, the logarithm equals 2, and the argument is \$2x + 4$.
2. Rewrite the equation \(\log_6 (2x + 4) = 2\) in exponential form: \$6^2 = 2x + 4$.
3. Calculate \$6^2\( (which is \)6\( raised to the power of \)2\() to simplify the right side of the equation, but do not finalize the numeric value; just write it as \)36$ for clarity.
4. Set up the equation \$36 = 2x + 4\( and isolate the variable term by subtracting 4 from both sides: \)36 - 4 = 2x$.
5. Solve for \(x\) by dividing both sides of the equation by 2: \(\frac{36 - 4}{2} = x\). This gives the exact solution for \(x\).

Key Concepts

Properties of Logarithms

Understanding the properties of logarithms, such as the definition log_b(a) = c means b^c = a, is essential. This allows converting logarithmic equations into exponential form to solve for the variable.

Solving Exponential Equations

Once the logarithmic equation is rewritten in exponential form, solving for the variable involves algebraic manipulation, such as isolating the variable and simplifying expressions.

Domain Restrictions of Logarithmic Functions

Logarithmic functions are only defined for positive arguments. When solving, it is important to check that the solutions make the argument inside the log positive to ensure valid answers.