Problem 61 - Algebra 2

Question

Show that each statement is true. \(\log _{16} 2 \cdot \log _{2} 16=1\)

Step-by-Step Solution

Answer

The statement is true: \( \log_{16} 2 \cdot \log_{2} 16 = 1 \).

Step 1: Convert to Common Base

First, we express each logarithm with a base that is easy to work with. Note that \[ \log_{16} 2 = \frac{\log_{2} 2}{\log_{2} 16} \]and\[ \log_{2} 16 = \frac{\log_{2} 16}{\log_{2} 2} \]However, to simplify easily, we use the relationship that bases are powers of each other.

Step 2: Simplify Each Logarithm

Recognize that 16 is a power of 2, specifically, \[ 16 = 2^4 \].Then \[ \log_{2} 16 = \log_{2} (2^4) = 4 \].Similarly, \[ \log_{16} 2 = \frac{1}{\log_{2} 16} = \frac{1}{4} \]by using the change of base formula and the relationship that \( \log_{b} a = \frac{1}{\log_{a} b}\).

Step 3: Calculate the Product

Now, substitute the simplified values:\[ \log_{16} 2 \cdot \log_{2} 16 = \frac{1}{4} \cdot 4 \].Multiply the two fractions:\[ \frac{1}{4} \cdot 4 = 1 \].

Step 4: Conclusion

Since the product of \( \log_{16} 2 \) and \( \log_{2} 16 \) simplifies to 1, it confirms the statement is true. Hence,\[ \log_{16} 2 \cdot \log_{2} 16 = 1 \].

Key Concepts

Change of Base Formula

The change of base formula is a fundamental tool when dealing with logarithms of different bases. It allows one to rewrite a logarithm in terms of common bases like 10 or e (natural logarithms), which are usually more convenient. Mathematically, for two numbers, a base \( b \) and the number \( a \), it is expressed as:

• \( \log_b a = \frac{\log_k a}{\log_k b} \)

Here \( k \) is a new base that you choose. This is particularly useful when the original base isn't directly compatible with the number you’re taking a logarithm of.
When you look at the problem involving \( \log_{16} 2 \) and \( \log_{2} 16 \), the change of base formula simplifies the calculations by turning them into expressions involving the same base. Using base 2 in the step by step solution helps us simplify much quicker. Understanding this formula is crucial in solving complex logarithmic equations like the one in the exercise.

Logarithmic Identities

Logarithmic identities are relationships that express logarithmic functions in different ways, enabling their simplification or expansion. Some foundational identities to keep in mind include:

• \( \log_b 1 = 0 \)
• \( \log_b b = 1 \)
• \( \log_b (xy) = \log_b x + \log_b y \)
• \( \log_b \left( \frac{x}{y} \right) = \log_b x - \log_b y \)
• \( \log_b (x^r) = r \cdot \log_b x \)

In the provided exercise, special attention is given to the identity \( \log_{b} a = \frac{1}{\log_{a} b} \). This identity highlights a symmetric property of logarithms, meaning swapping the base and the number reverses the reciprocal property. By using the identity in the problem, \( \log_{16} 2 \) simplifies to a convenient reciprocal form, confirming the solution quickly.

Exponents in Logarithms

Exponents play a critical role in understanding and solving logarithmic functions. When a number is expressed in terms of powers, it directly relates to logarithms because the latter essentially ask the question: "to what power must the base be raised to obtain a given number?"
In the original problem, we see this with the realization that 16 is expressed as \( 2^4 \). Recognizing power relationships like this simplifies solving logarithms.

• \( \log_{2} 16 = \log_{2} (2^4) = 4 \)

Expressing numbers as exponents is often an initial step in simplifying logarithmic expressions, making them more straightforward to handle. This is vital in both manual calculations and automated solving processes. Understanding how exponents and logarithms interact provides clarity into the mechanics of logarithmic functions and greatly aids in problem-solving.