Solve each equation. x = 3^log3 8

Verified step by step guidance

1. Recognize that the equation is in the form \( x = 3^{\log_3 8} \).
2. Understand that \( 3^{\log_3 8} \) is an example of an exponential expression where the base of the exponent and the base of the logarithm are the same.
3. Apply the property of logarithms and exponents: \( a^{\log_a b} = b \).
4. Use this property to simplify the expression: \( 3^{\log_3 8} = 8 \).
5. Conclude that \( x = 8 \).

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. The logarithm log_a(c) answers the question: 'To what exponent must the base a be raised to produce c?' Understanding logarithms is essential for manipulating and solving equations involving exponential terms.

Properties of Logarithms

The properties of logarithms, such as the product, quotient, and power rules, provide tools for simplifying logarithmic expressions. For instance, log_a(b * c) = log_a(b) + log_a(c) and log_a(b^n) = n * log_a(b). These properties are crucial for breaking down complex logarithmic equations into more manageable parts.

Exponential Functions

Exponential functions are mathematical expressions in the form f(x) = a * b^x, where a is a constant, b is the base, and x is the exponent. They exhibit rapid growth or decay and are fundamental in various applications, including finance and population studies. Understanding how to manipulate and solve equations involving exponential functions is key to solving the given problem.