Problem 53 - Algebra 2

Question

Express each logarithm in terms of common logarithms. Then approximate its value to four decimal places. $$ \log _{5} 8 $$

Step-by-Step Solution

Answer

\( \log_{5} 8 \approx 1.2920 \)

Step 1: Convert to Common Logarithms

To express \( \log_{5} 8 \) in terms of common logarithms, we use the change of base formula: \( \log_{b} a = \frac{\log_{10} a}{\log_{10} b} \). This gives us: \[ \log_{5} 8 = \frac{\log_{10} 8}{\log_{10} 5} \]

Step 2: Calculate the Common Logarithms

Next, we calculate \( \log_{10} 8 \) and \( \log_{10} 5 \) using a calculator:- \( \log_{10} 8 \approx 0.9031 \).- \( \log_{10} 5 \approx 0.6990 \).

Step 3: Solve Using Values

Substitute the approximated values of \( \log_{10} 8 \) and \( \log_{10} 5 \) into the expression:\[ \log_{5} 8 = \frac{0.9031}{0.6990} \]

Step 4: Perform the Division

Finally, divide 0.9031 by 0.6990 to approximate the logarithm's value:\[ \log_{5} 8 \approx 1.2920 \]

Key Concepts

Change of Base Formula

The change of base formula is a very useful tool in logarithms that allows us to convert logarithms of any base into a base that is more easily managed, often base 10 (the common logarithm) or base e (the natural logarithm). The formula is expressed as:

• \( \log_{b} a = \frac{\log_{10} a}{\log_{10} b} \)

This formula transforms a logarithm with an arbitrary base \( b \) into the ratio of logarithms that can be calculated using a scientific calculator, which typically only handles base 10 and base e.
In many problems involving logarithms, you may want to use the change of base formula when:

• You have a calculator that only supports base 10 or base e.
• You need to simplify logarithmic expressions.

This formula can be particularly helpful when you need to approximate the value of logarithms, especially when the direct computation isn't feasible.

Common Logarithm

A common logarithm is a logarithm to the base 10, often denoted as \( \log \) or \( \log_{10} \). It's one of the most frequently used logarithmic bases, primarily because our numeral system is base 10, and calculators are equipped to easily compute these values.
When you see \( \log a \) without an explicitly stated base, it's typically assumed to mean \( \log_{10} a \). Common logarithms have many applications across various scientific fields and are integral in solving logarithmic problems where base 10 simplifies calculations.
Key points about common logarithms include:

• They are especially useful when dealing with powers of ten.
• Calculators can effortlessly provide precise values for any numerical value using the common logarithm.
• Engineers and scientists frequently rely on them due to their alignment with our base-10 system.

Understanding common logarithms allows you to transition among different bases, which aids in finding approximate values in mathematical equations.

Logarithmic Expression Approximation

Approximating the value of a logarithmic expression involves using a calculator or logarithm tables to determine numerical values for logarithms that can't be easily worked out by hand. Since most real-world problems don't result in neat, round numbers, approximating is often necessary.
To approximate a logarithmic expression, decode it using the change of base formula to convert it into a base your calculator can handle, such as base 10. For example, given \( \log_{5} 8 \), you would change it to:

• \( \log_{5} 8 = \frac{\log_{10} 8}{\log_{10} 5} \)

After this transformation, you compute each component:

• Calculate \( \log_{10} 8 \approx 0.9031 \).
• Calculate \( \log_{10} 5 \approx 0.6990 \).

Finally, finish by dividing these results to approximate the original log:

• \( \log_{5} 8 \approx 1.2920 \).

Approximation is a crucial skill in mathematics as it deals with real numbers, providing a means to interpret data with a degree of accuracy that complements raw calculation plus intuition.