Evaluate each expression without using a calculator. log₇ √7

Verified step by step guidance

1. Recognize that the expression is \( \log_7 \sqrt{7} \), which means the logarithm base 7 of the square root of 7.
2. Rewrite the square root of 7 using an exponent: \( \sqrt{7} = 7^{\frac{1}{2}} \).
3. Substitute this back into the logarithm: \( \log_7 7^{\frac{1}{2}} \).
4. Use the logarithm power rule, which states \( \log_b (a^c) = c \cdot \log_b a \), to simplify: \( \log_7 7^{\frac{1}{2}} = \frac{1}{2} \cdot \log_7 7 \).
5. Since \( \log_7 7 = 1 \) (because any log base of itself is 1), the expression simplifies to \( \frac{1}{2} \cdot 1 = \frac{1}{2} \).

Key Concepts

Properties of Logarithms

Logarithms have specific properties that simplify expressions, such as the product, quotient, and power rules. For example, the power rule states that log_b(a^c) = c * log_b(a), which helps in rewriting and evaluating logarithmic expressions without a calculator.

Understanding Radicals as Exponents

A square root can be expressed as an exponent of 1/2, so √7 is equivalent to 7^(1/2). This conversion allows the use of exponent rules within logarithmic expressions, making it easier to simplify and evaluate the expression.

Logarithm of the Base

The logarithm of a base to itself is always 1, meaning log_b(b) = 1. This fundamental fact is essential when simplifying expressions like log_7(7^(1/2)), as it directly leads to the evaluation of the logarithm.