Problem 26 - Precalculus with Limits

Question

Using Properties of Logarithms In Exercises \(21-36,\) find the exact value of the logarithmic expression without using a calculator. (If this is not possible, then state the reason.) $$\log _{4} 16^{2}$$

Step-by-Step Solution

Answer

The value of the logarithmic expression \( \log _{4} 16^{2} \) is 4

Step 1 - Breaking down the expression using the power rule of logarithms

We start by applying the power rule of logarithms, which allows us to move the exponent in the argument out to the front of the logarithm. Thus, \( \log _{4} 16^{2} \) can be written as \( 2 * \log _{4} 16 \)

Step 2 - Calculating the value of the simplified logarithm

Now we have to calculate the value of \( \log _{4} 16 \), which is asking '4 to the power of what equals 16'. It is evident that \( 4^{2} = 16 \), hence the value of \( \log _{4} 16 \) will be 2

Step 3 - Final computation

As per step 1, our given problem can be written as \( 2 * \log _{4} 16 \), now that we have computed \( \log _{4} 16 \) to be 2, substitute the value in equation, so we get \( 2 * 2 = 4 \) this is the final required solution

Key Concepts

Power Rule of Logarithms

Understanding the power rule of logarithms is essential for simplifying complex logarithmic expressions. This rule states that a logarithm of a number raised to a power, such as \( \text{log}_b(m^n) \), can be rewritten as the power multiplied by the logarithm of the base number, formalized as \( n \times \text{log}_b(m) \).

For example, let's consider the expression \( \text{log}_{4}(16^2) \). To simplify this expression using the power rule, we can 'pull' the exponent of 2 in front of the logarithm, resulting in \( 2 \times \text{log}_{4}(16) \). This simplification step reduces the complexity significantly since now we only need to evaluate \( \text{log}_{4}(16) \), which is a much simpler task.

Logarithmic Expressions

Logarithmic expressions translate the operation of exponentiation into multiplication and division, allowing for easier manipulation. A logarithm is essentially the reverse of exponentiation and asks the following question: 'To what exponent must we raise the base to get the given number?'.

For instance, if we consider the base 4 and number 16, the logarithmic expression \( \text{log}_4(16) \) is asking 'What power must we raise 4 to get 16?'. Since \( 4^2 = 16 \), we know that the answer is 2. Logarithms provide a way to work with growth patterns, decay rates, and many phenomena that involve multiplicative change in various scientific, mathematical, and real-world contexts.

Simplifying Logarithms

Simplifying logarithms makes them more digestible for computation and understanding. This process often involves using different logarithmic properties like the power rule, product rule, and quotient rule. The goal is to break down complex logarithmic expressions into simpler, more manageable forms.

Returning to our example, \( \text{log}_{4}(16) \) is simplified to 2 because we recognize that 16 is a power of the base 4. Similarly, other complex expressions can be simplified by identifying and utilizing the relationships between the numbers and the base. This process doesn't just make it possible to calculate values without a calculator, but also helps in developing a clear understanding of exponential and logarithmic relationships.