Solving equations Solve the following equations.


log₈ x = 1/3

Verified step by step guidance

1. First, understand that the equation \( \log_8 x = \frac{1}{3} \) is asking for the value of \( x \) such that when 8 is raised to the power of \( \frac{1}{3} \), it equals \( x \).
2. Rewrite the logarithmic equation in its exponential form: \( x = 8^{\frac{1}{3}} \).
3. Recognize that raising a number to the power of \( \frac{1}{3} \) is equivalent to taking the cube root of that number.
4. Thus, \( x = \sqrt[3]{8} \).
5. Finally, simplify the expression \( \sqrt[3]{8} \) to find the value of \( x \).

Key Concepts

Logarithms

Logarithms are the inverse operations of exponentiation, allowing us to solve for the exponent in equations of the form a^b = c. In the equation log₈ x = 1/3, it indicates that 8 raised to the power of 1/3 equals x. Understanding logarithmic properties is essential for manipulating and solving such equations.

Change of Base Formula

The Change of Base Formula allows us to convert logarithms from one base to another, which is particularly useful when dealing with bases that are not easily computable. The formula states that logₐ b = logₓ b / logₓ a for any positive x. This can simplify calculations, especially when using calculators that typically only compute base 10 or base e logarithms.

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 the relationship between logarithms and exponents. In the context of the given equation, recognizing that solving log₈ x = 1/3 involves rewriting it as an exponential equation helps in finding the value of x directly.