Problem 38 - Algebra 2 and Trigonometry

Question

In \(27-56,\) evaluate each logarithmic expression. Show all work. $$ 16 \log _{4} \frac{1}{64} $$

Step-by-Step Solution

Answer

The evaluated expression is -48.

Step 1: Apply the logarithm property

We start by applying the property of logarithms: \[\log_b(\frac{1}{a}) = -\log_b(a)\]This property states that the logarithm of a reciprocal is the negative logarithm of the original number. So, we can rewrite the expression:\[\log_4\left(\frac{1}{64}\right) = -\log_4(64)\]

Step 2: Express 64 as a power of 4

Next, we need to represent 64 as a power of 4. Since 64 is equal to \(4^3\), we use this to rewrite the expression:\[-\log_4(64) = -\log_4(4^3)\]

Step 3: Use the power rule of logarithms

We now apply the power rule of logarithms, which states: \[\log_b(a^n) = n \cdot \log_b(a)\]Applying this rule, we simplify:\[-\log_4(4^3) = -3 \cdot \log_4(4)\]

Step 4: Evaluate \(\log_4(4)\)

Since the base and the number are identical in \(\log_4(4)\), it equals 1 because any base logarithm of the same base is always 1. Thus:\[\log_4(4) = 1\]Substitute back to get:\[-3 \cdot 1 = -3\]

Step 5: Multiply by 16

Finally, we multiply the result by 16 as required by the problem:\[16 \times (-3) = -48\]

Key Concepts

Logarithm Properties

Logarithms are mathematical tools that help articulate the power to which a number, called the base, must be raised to produce another number.
One significant property of logarithms is that they simplify complex equations, especially those involving exponents. Here's a handy property of logarithms we used in the exercise:

• Reciprocal Property: The logarithm of a reciprocal can be transformed to the negative logarithm of the original number, which is depicted as \( \log_b(\frac{1}{a}) = -\log_b(a) \). This comes in handy when dealing with fractions or division within logarithms.
  

Applying these properties accurately is crucial in evaluating logarithmic expressions efficiently.

Logarithmic Expression Evaluation

Evaluating a logarithmic expression involves using properties and known values to break down the expression into something manageable. For instance, the expression \( \log_4(\frac{1}{64}) \) can be assessed as follows:

• Begin with the reciprocal property to reformulate the logarithm into a simpler form.
  
• Express any numbers as powers of their bases where possible, which aids in simplifying the logarithmic calculations. Here, noticing that 64 can be expressed as \( 4^3 \) is crucial because it simplifies the evaluation using further logarithm rules.
  

Practicing these steps ensures clarity and confidence in handling complex logarithmic expressions.

Power Rule of Logarithms

The power rule in logarithms is an incredibly useful tool that allows us to manage expressions involving exponents efficiently. Here's how it works:

• Power Rule: The power rule is expressed as \( \log_b(a^n) = n \cdot \log_b(a) \). This tells us that you can bring down the power in front of the logarithm, turning what could be a complex calculation into manageable arithmetic.
  

Using the power rule in our exercise changed \( -\log_4(4^3) \) into \( -3 \cdot \log_4(4) \), leveraging the simplicity when the base and the logarithmic number are the same. Always remember, the goal is to simplify where possible, using these logarithmic properties.