Problem 64 - Algebra 2

Question

Use \(\log _{7} 2 \approx 0.3562\) and \(\log _{7} 3 \approx 0.5646\) to approximate the value of each expression. $$ \log _{7} 27 $$

Step-by-Step Solution

Answer

\( \log_{7} 27 \approx 1.6938 \).

Step 1: Identify the Expression for Logarithm

We need to find \( \log_{7} 27 \). Notice that 27 can be expressed as \( 27 = 3^3 \). Therefore, \( \log_{7} 27 \) can be rewritten using the property of logarithms that states \( \log_b (a^n) = n \cdot \log_b a \).

Step 2: Rewrite the Logarithm

Using the property from Step 1, rewrite the expression: \[ \log_{7} 27 = \log_{7} (3^3) = 3 \cdot \log_{7} 3 \]

Step 3: Substitute Known Values

Now substitute the known value of \( \log_{7} 3 \) from the problem, which is approximately 0.5646, into the expression:\[ 3 \cdot \log_{7} 3 = 3 \cdot 0.5646 \]

Step 4: Calculate the Product

Calculate the product:\[ 3 \times 0.5646 = 1.6938 \]

Step 5: Conclusion

Therefore, the approximate value of \( \log_{7} 27 \) is 1.6938.

Key Concepts

Logarithmic Properties

Logarithmic properties are powerful tools for simplifying and solving expressions involving logs. One of the most important properties is the power rule, which states: if you have a logarithm of an exponent, such as

• \( \log_b (a^n) \)

this can be rewritten as

• \( n \cdot \log_b a \).

This rule helps break down complex expressions into simpler, more manageable parts. It turns a challenging multiplication problem into an easier multiplication problem.
Another critical property is the product rule, which converts the log of a product into the sum of logs:

• \( \log_b (xy) = \log_b x + \log_b y \).

Using these rules can simplify calculations and help us approximate logs without a calculator. For example, to approximate \( \log_7 27 \), we used the base rule above to rewrite it as \( 3 \cdot \log_7 3 \). Breaking down logarithmic expressions in this way makes them far more accessible.

Exponentiation

Exponentiation is a mathematical operation that involves raising a number, known as the base, to the power of an exponent. It expresses how many times the base is multiplied by itself. For example, in the expression

• \( 3^3 \)

3 is the base, and 3 is the exponent, implying

• \( 3 \times 3 \times 3 = 27 \).

Understanding how to express numbers as powers can simplify working with logs, as seen in the step of rewriting \( \log_7 27 \) as \( \log_7 (3^3) \). Using powers can help to apply the logarithmic properties effectively.
Recognizing expressions that can be rewritten using exponentiation is vital. It allows you to apply properties like the power rule to simplify logarithms and makes calculations more straightforward.

Base 7 Logarithms

A base 7 logarithm is simply a logarithm that uses 7 as its base. This means that when you see an expression like \( \log_7 x \), you're calculating what power 7 needs to be raised to produce \( x \).
Calculating base 7 logs without a calculator usually requires breaking down the number into factors that involve the base, as we did with the expression \( \log_7 27 = \log_7 (3^3) \). By expressing 27 in terms of powers of other known numbers and using the base we have, calculations become easier.
For example, knowing \( \log_7 2 \) and \( \log_7 3 \), you can determine more complex logs by evaluating simple arithmetic operations, such as multiplication of the known values.

• If \( x = 3 \times 3 \times 3 \), then \( \log_7 x = 3 \cdot \log_7 3 \).

Understanding the nature of base 7 logarithms helps solve equations involving these logs efficiently, especially when combined with logarithmic properties.