Evaluate each expression without using a calculator. log₆₄ 8

Verified step by step guidance

1. Recognize that the expression \( \log_{64} 8 \) asks for the exponent \( x \) such that \( 64^x = 8 \).
2. Express both 64 and 8 as powers of the same base. Since both are powers of 2, write \( 64 = 2^6 \) and \( 8 = 2^3 \).
3. Rewrite the equation \( 64^x = 8 \) as \( (2^6)^x = 2^3 \).
4. Use the power of a power property to simplify the left side: \( 2^{6x} = 2^3 \).
5. Since the bases are the same, set the exponents equal: \( 6x = 3 \), then solve for \( x \) by dividing both sides by 6.

Key Concepts

Logarithm Definition

A logarithm answers the question: to what exponent must the base be raised to produce a given number? For example, log_b(a) = c means b^c = a. Understanding this definition is essential to evaluate logarithmic expressions.

Change of Base and Prime Factorization

To evaluate logarithms without a calculator, express both the base and the argument as powers of the same prime number. This allows rewriting the logarithm in terms of simpler exponents, facilitating easier calculation.

Properties of Logarithms

Logarithm properties, such as log_b(m^n) = n*log_b(m), help simplify expressions. Applying these rules can break down complex logarithms into manageable parts, making evaluation straightforward.