Problem 19 - Algebra 2

Question

Use \(\log _{5} 2 \approx 0.4307\) and \(\log _{5} 3 \approx 0.6826\) to approximate the value of each expression. \(\log _{5} 8\)

Step-by-Step Solution

Answer

\( \log_{5} 8 \approx 1.2921 \) using the power rule and given values.

Step 1: Express 8 Using Known Logarithms

To approximate \( \log_{5} 8 \), start by expressing the number 8 in terms of integers whose logarithms are known. Recognize that 8 can be rewritten as \( 2^3 \). Thus, \( \log_{5} 8 = \log_{5} (2^3) \).

Step 2: Apply the Power Rule of Logarithms

Use the power rule of logarithms which states \( \log_{b} (a^n) = n \cdot \log_{b} a \). Apply this rule to \( \log_{5} (2^3) \):\[ \log_{5} (2^3) = 3 \cdot \log_{5} 2 \] Substitute \( \log_{5} 2 \approx 0.4307 \) into the equation.

Step 3: Perform the Multiplication

Now calculate the expression:\[ 3 \cdot \log_{5} 2 \approx 3 \cdot 0.4307 \]Perform the multiplication to find:\[ 3 \cdot 0.4307 \approx 1.2921 \]

Step 4: Final Answer

Therefore, the approximate value of \( \log_{5} 8 \) is \( 1.2921 \).

Key Concepts

Power Rule of Logarithms

The power rule of logarithms is a handy tool whenever you need to calculate the logarithm of a number raised to a power. This rule states that for any positive number \(a\), base \(b\), and exponent \(n\), the logarithm \(\log_{b} (a^n)\) can be rewritten as \(n \cdot \log_{b} a\). This means you can "bring down" the exponent \(n\) as a multiplier in front of the logarithm.

For example, in the expression \(\log_{5} (2^3)\), using the power rule simplifies this to \(3 \cdot \log_{5} 2\). Notice how 3, which is the exponent of 2, is placed as a coefficient before the logarithm.

• It simplifies complex logarithmic calculations.
• Allows multiplication instead of exponentiation, making calculations more manageable.

Understanding this step is crucial, as it greatly reduces the time and effort needed to solve logarithmic expressions involving exponents.

Approximation in Logarithmic Calculations

Following the power rule, the next crucial step is performing approximations in logarithmic calculations when exact values are not possible or practical.

Approximations allow us to work with numbers like \( \log_{5} 2 \approx 0.4307 \) when only an approximate value is available. These values can be used effectively to still get a meaningful result in bigger calculations. We used the approximate value of \( \log_{5} 2 \) to find the value of \( \log_{5} 8 \).

Keep in mind:

• Approximations help avoid cumbersome calculations with high precision, which might be unnecessary for the context.
• The result is usually close enough for practical purposes, maintaining a balance between exactness and ease of computation.

Therefore, being able to effectively use approximations can be a real time-saver in many mathematical settings.

Understanding Exponents

Exponents are a mathematical notation indicating the number of times a number (known as the base) is to be multiplied by itself. In our problem, we expressed 8 as \(2^3\). Here, 2 is the base and 3 is the exponent, meaning 2 is multiplied by itself three times: \(2 \times 2 \times 2 = 8\).

Error-free manipulation and understanding of exponents is crucial:

• The base and exponent structure simplifies large number handling and calculations involving repeated multiplication.
• In logarithms, recognizing the repeated base helps reframe the problem using the power rule.

Proper use of exponents enables complex problems to be converted into simpler forms, whether solving equations or dealing with logarithms. This foundation of exponent understanding is key when applying rules like the power rule of logarithms.