Solving Logarithmic Equations: log₆x = 2

Q1. Solve the following logarithmic equation: log₆x = 2

Background

Topic: Logarithmic and Exponential Equations

This question tests your understanding of how to solve logarithmic equations by converting them to exponential form. This is a fundamental skill in precalculus, especially when working with logarithms and their properties.

Key Terms and Formulas

• Logarithmic Equation: An equation involving a logarithm with a variable inside its argument.

• Exponential Form: $\log_a x = b \iff x = a^b$

• Domain: The argument of the logarithm must be positive ($x > 0$).

Step-by-Step Guidance

1. Identify the base of the logarithm. Here, the base is 6.

2. Recall the definition: $\log_6 x = 2$ means $x$ is the number such that $6$raised to the power $2$ equals $x$.

3. Rewrite the logarithmic equation in exponential form: $x = 6^2$.

4. Check the domain: Make sure $x > 0$ (which will be satisfied in this case).

Try solving on your own before revealing the answer!

Final Answer: x = 36

$x = 6^2 = 36$

We converted the logarithmic equation to exponential form and solved for $x$. The solution satisfies the domain requirement ($x > 0$).