Problem 60 - Algebra 2

Question

Write true or false for each statement. Justify your answer. \(\log _{3} 8=3 \log _{3} 2\)

Step-by-Step Solution

Answer

False, the statement \( \log_3 8 = 3 \log_3 2 \) is not correct, since they result in different values when evaluated.

Step 1: Understanding Logarithm Properties

One of the key properties of logarithms states that \( \log_b(m^n) = n \cdot \log_b(m) \). Testing the given statement requires you to apply this property.

Step 2: Evaluating the Left Side of the Equation

Evaluate the left-hand side of the equation. \( \log_3 8 \) does not simplify directly as 8 is not a natural power of 3. Using the change-of-base formula, we can express it in terms of natural logarithms, yielding \( \frac{\ln(8)}{\ln(3)} \).

Step 3: Evaluating the Right Side of the Equation

Evaluate the right-hand side of the equation. \( 3 \log_3 2 \) simplifies to 3, as \( \log_3 2 \) is the power to which 3 must be raised to yield 2.

Step 4: Comparing the Results

Now, compare the values from step 2 and 3, i.e., compare \( \frac{\ln(8)}{\ln(3)} \) and 3. As these are not equal, the statement is not correct.

Key Concepts

Understanding Change-of-Base Formula

The change-of-base formula is an essential tool for evaluating logarithms that cannot be easily simplified using the available base. This formula allows us to rewrite logarithms in terms of other bases, such as the natural logarithm (base \( e \)) or common logarithm (base 10), which are more convenient for calculations. The change-of-base formula is given by: \[\log_b(a) = \frac{\log_c(a)}{\log_c(b)}\] Here, \( b \) is the base of the original logarithm, \( a \) is the argument, and \( c \) is the new base. You can choose any base \( c \), but natural and common logarithms are frequently used because they can be directly calculated with most calculators.

• The change-of-base formula is especially useful when dealing with logarithms with uncommon bases, such as \( \log_3 8 \) in our example.
• It makes it possible to compute logarithms with any base without requiring detailed knowledge of the base itself.

For instance, if you don't have a calculator that directly supports base-3 logarithms, expressing \( \log_3 8 \) as \( \frac{\ln(8)}{\ln(3)} \) is a practical approach. This allows you to solve the logarithmic expression using familiar bases.

Solving Logarithmic Equations

A logarithmic equation involves an unknown variable inside a logarithm. Solving these equations typically requires the application of properties of logarithms and sometimes the change-of-base formula. Here is a brief guide to handling these calculations:

• First, familiarize yourself with logarithmic properties, such as the product, quotient, and power rules.
• Next, simplify the logarithmic equation as much as possible. For example, expressing each logarithm in a common base could simplify the comparison of both sides.
• In our example, \( \log_3 8 = 3 \log_3 2 \) simplifies by using the power rule: \( \log_b(m^n) = n \cdot \log_b(m) \), testing if one side can be expressed as a power of the other.

If the expressions after simplification don't match, the equation is not true. In our original problem, applying these methods helped demonstrate that the two sides were not equal, confirming that the statement is false.

Evaluating Logarithmic Expressions

Evaluating a logarithmic expression means finding its numerical value. When directly evaluating logs like \( \log_3 8 \), we notice that 8 is not a pure power of 3, making it difficult to simplify directly. Here's how you can evaluate such expressions:

• Use the change-of-base formula to convert it into a more manageable form as demonstrated previously, such as \( \frac{\ln(8)}{\ln(3)} \), for easier calculation with a calculator.
• Check if the argument of the log is a power of the base. In our example, if 8 were a power of 3, simplifying directly would be feasible.

By accurately applying these approaches, students can calculate complex logs, making these tools invaluable for solving both simple and intricate logarithmic equations. Expressing \( \log_3 8 \) in a more user-friendly form with the formula and verifying against known solutions can ensure clarity and understanding.