Problem 37 - College Algebra

Question

Evaluate each expression without using a calculator. $$8^{\log _{2} 19}$$

Step-by-Step Solution

Answer

The value of the expression \(8^{log_{2} 19}\) is \(19\).

Step 1: Understanding the change of base property

The change of base formula states that \(a^{log_b c} = c\), given that \(a\), \(b\), and \(c\) are positive real numbers and \(a ≠ 1\). This is because the logarithm operation \(log_b c\) means to what power we must raise \(b\) to get \(c\), and that power is then used as the exponent of \(a\). When \(a\) and \(b\) are the same, they effectively cancel each other out, leaving only \(c\).

Step 2: Applying the change of base property

By using the change of base property on the given expression \(8^{log_{2} 19}\), we replace \(a\) with \(2^3\) (since \(8\) is \(2^3\)), \(b\) with \(2\), and \(c\) with \(19\). This leads to the expression \(2^{3\cdot log_{2} 19}\).

Step 3: Simplifying the expression

Since the values of \(a\) and \(b\) are now the same, they cancel each other out based on the change of base property. This simplifies the expression to just \(19\).

Key Concepts

Logarithms

Logarithms are fundamental tools in mathematics, essential for solving problems involving exponential relationships. They act as the inverse operation of exponentiation, meaning they 'undo' what exponents do. A logarithm answers the question: To what power must we raise a certain base to obtain a specific number? The general form is written as \( \log_b(c) \), with \( b \) as the base and \( c \) as the number we want to match by raising \( b \) to a power.

For example, the expression \( \log_2(8) \) is asking, 'To what power must we raise 2 to get 8?' The answer is 3, because \( 2^3 = 8 \). This relationship is pivotal in connecting logarithmic and exponential forms, allowing us to convert between the two for easier calculation, particularly when evaluating expressions without a calculator.

Exponentiation

Exponentiation is an arithmetic operation that involves raising a number, called the base, to a power, which is called the exponent. It is denoted as \( b^n \), where \( b \) is the base and \( n \) is the exponent. This operation tells you to multiply the base by itself as many times as specified by the exponent.

For instance, in the expression \( 2^3 \), the number 2 is the base and 3 is the exponent, indicating that 2 should be multiplied by itself twice more (for a total of three times), giving us the result of 8. Exponentiation is the operation being inverted when we perform a logarithm. Understanding the direct relationship between exponentiation and logarithms is crucial, especially when working with expressions involving both operations.

Evaluating Expressions Without a Calculator

Evaluating expressions without a calculator is a valuable skill that strengthens your understanding of mathematical concepts and improves your mental calculation abilities. To evaluate expressions like \( 8^{\log_{2} 19} \) without a calculator, we use properties and relationships between operations as shortcuts. Here, the change of base formula is employed, allowing us to simplify complex expressions down to their most basic form.

Utilizing this formula, along with recognizing patterns and applying fundamental arithmetic principles, we manage to find accurate results through logical steps. For instance, understanding that \( 8 \) is \( 2^3 \) allows us to rewrite the original problem in a more manageable form, using exponentiation and logarithmic properties to simplify and ultimately find that the expression equals 19.