Problem 14 - Precalculus with Calculus Previews

Question

In Problems \(13-18\), find the exact value of the given logarithm. $$ \log _{4} 64 $$

Step-by-Step Solution

Answer

The exact value of \( \log_{4} 64 \) is 3.

Step 1: Understanding the Logarithm

The expression \( \log_{4} 64 \) asks us to determine which power we need to raise the base 4 to get the number 64. In other words, we need to solve for \( x \) in the equation \( 4^x = 64 \).

Step 2: Recognize 64 as a Power of 4

Observe that 64 is a power of 4. To find this, we rewrite 64 using powers of 4 and see if it equals any. We start by guessing and checking small integral powers: - \( 4^1 = 4 \)- \( 4^2 = 16 \)- \( 4^3 = 64 \)We notice that \( 4^3 = 64 \), confirming that 64 is \( 4 \) raised to the power of 3.

Step 3: Solve for the Exponent

Since \( 4^3 = 64 \), it follows that \( x = 3 \). Therefore, \( \log_{4} 64 = 3 \).

Step 4: Verify the Solution

Finally, verify by substituting back into the original equation: - Check if raising 4 to this power indeed gives 64:- \( 4^3 = 4 \times 4 \times 4 = 16 \times 4 = 64 \)The computation holds, so the solution is verified to be correct.

Key Concepts

Base of Logarithms

The base of a logarithm is an essential part of understanding how logarithms function. In the expression \( \log_{4} 64 \), the number 4 represents the 'base' of the logarithm.

• The base determines the number that is raised to a power to produce the given number.
• In this example, the question is: "To what power must we raise 4 to obtain 64?"

The purpose of the base is to serve as a reference point for the calculation. Changing the base changes the entire equation.Knowing how to manipulate and interpret the base is crucial when working with logarithms. For instance, the equation \( 4^x = 64 \) is the exponential form of the logarithmic equation. Understanding the base allows you to switch between these forms, which can simplify solving logarithmic problems.

Exponents

Exponents are vital when dealing with logarithms, as logarithms are essentially inverse operations to exponentiation. When solving \( \log_{4} 64 \), we are trying to figure out the exponent \( x \) in the expression \( 4^x = 64 \).

• Exponents tell us how many times to multiply the base by itself.
• For example, \( 4^3 \) means 4 multiplied by itself 3 times, resulting in 64.

Understanding exponents involves recognizing patterns in numbers, such as how quickly they can grow as the exponent increases. Working with exponents means having a grasp of both positive and negative values, as well as fractional exponents. In solving logarithms like \( \log_{4} 64 \), mastery of these concepts is necessary to reach accurate solutions.

Powers of Numbers

Finding powers of numbers is a foundational skill in mathematics and is critical for understanding logarithms. When working out \( \log_{4} 64 \), the task involves identifying what power of 4 equals 64.

• Powers of numbers are the result of multiplying a number by itself a certain number of times.
• In this case, we calculate powers of 4 to see which one results in 64.
• After computation, we find \( 4^3 = 64 \), indicating 4 must be raised to the power of 3.

To effectively find a power, you can use both strategies of guessing and checking or employing algebraic techniques. By practicing these strategies, you can improve your ability to quickly identify powers, making logarithm problems easier to solve.