Evaluate each expression without using a calculator. $8^{\(\log\)_8 19}$

Verified step by step guidance

1. Recognize that the expression is of the form \(a^{\log_a b}\), where the base of the exponent and the base of the logarithm are the same (in this case, 8).
2. Recall the logarithmic identity: \(a^{\log_a b} = b\). This identity holds because the logarithm \(\log_a b\) is the exponent to which you raise \(a\) to get \(b\).
3. Apply this identity directly to the expression \(8^{\log_8 19}\), which simplifies to just 19.
4. Understand that this simplification works without any calculation because the logarithm and the exponent base cancel each other out.
5. Therefore, the value of the expression \(8^{\log_8 19}\) is simply 19.

Key Concepts

Logarithmic and Exponential Functions

Logarithms and exponents are inverse operations. The logarithm log_b(a) answers the question: to what power must the base b be raised to get a? Understanding this inverse relationship is key to simplifying expressions like b^(log_b(x)).

Properties of Logarithms and Exponents

One important property is that b^(log_b(x)) = x, where b is the base of both the exponent and the logarithm. This property allows simplification of expressions where the base of the exponent matches the base of the logarithm.

Evaluating Expressions Without a Calculator

When evaluating expressions like 8^(log_8 19) without a calculator, use algebraic properties to simplify rather than compute directly. Recognizing patterns and applying properties reduces complex expressions to simpler forms.