Solve each equation. log￬4 x = 3

Verified step by step guidance

1. Understand that the equation \( \log_4 x = 3 \) is asking for the value of \( x \) such that when 4 is raised to the power of 3, it equals \( x \).
2. Rewrite the logarithmic equation in its exponential form: \( x = 4^3 \).
3. Calculate \( 4^3 \) by multiplying 4 by itself three times: \( 4 \times 4 \times 4 \).
4. Simplify the expression to find the value of \( x \).
5. Verify your solution by substituting \( x \) back into the original equation to ensure that \( \log_4 x = 3 \) holds true.

Key Concepts

Logarithms

A logarithm is the inverse operation to exponentiation, representing the power to which a base must be raised to obtain a given number. In the equation log₄ x = 3, it indicates that 4 raised to the power of 3 equals x. Understanding logarithms is essential for solving equations involving them.

Exponential Functions

Exponential functions are mathematical functions of the form f(x) = a^x, where 'a' is a positive constant. They are crucial for understanding how logarithms work, as they describe the relationship between the base and the exponent. In this case, knowing that 4^3 = 64 helps in solving the logarithmic equation.

Change of Base Formula

The change of base formula allows the conversion of logarithms from one base to another, facilitating easier calculations. It states that logₐ b = logₓ b / logₓ a for any positive x. This concept can be useful if you need to evaluate logarithms with bases that are not easily computable.