Problem 37 - Algebra 2

Question

Express each logarithm in terms of common logarithms. Then approximate its value to four decimal places. $$ \log _{3} 8 $$

Step-by-Step Solution

Answer

\( \log_{3} 8 \approx 1.8928 \)

Step 1: Use the Change of Base Formula

For logarithms, the change of base formula is very useful. It allows us to express a logarithm with any base in terms of common logarithms (base 10) or natural logarithms (base e). The formula is:\[\log_{b} a = \frac{\log_{10} a}{\log_{10} b}\]In this exercise, you need to write \( \log_{3} 8 \) using base 10 logarithms.

Step 2: Apply the Formula to the Given Logarithm

Using the change of base formula, express \( \log_{3} 8 \) using common logarithms:\[\log_{3} 8 = \frac{\log_{10} 8}{\log_{10} 3}\]Now, the task is to compute \( \log_{10} 8 \) and \( \log_{10} 3 \).

Step 3: Calculate the Values with a Calculator

Use a calculator to find the numerical values of the common logarithms:- \( \log_{10} 8 \approx 0.9031 \)- \( \log_{10} 3 \approx 0.4771 \)Now substitute these values back into the expression:

Step 4: Compute the Result

Substituting the values back into the formula:\[\log_{3} 8 = \frac{0.9031}{0.4771}\approx 1.8928\]This is the decimal approximation of \( \log_{3} 8 \) rounded to four decimal places.

Key Concepts

Common Logarithms

Common logarithms are logarithms with base 10, often represented as \( \log_{10} \). They are widely used in mathematics, especially in scientific calculations. The reason for this is that our number system is based on 10, which makes base 10 logarithms very convenient.

When you see \( \log \) without a specified base, it usually means base 10. The common logarithm of a number \( x \) is the power to which 10 must be raised to yield \( x \). For example, \( \log_{10} 100 = 2 \) because \( 10^2 = 100 \).

Using calculators or logarithm tables, you can easily find the value of common logarithms. This makes them a powerful tool to express logarithms of different bases in a more manageable form, as seen in our task of expressing \( \log_{3} 8 \).

Base Conversion

Base conversion is essential when dealing with logarithms of different bases. The change of base formula helps in converting any logarithm to a base that is easier to work with, like base 10 or base \( e \) (natural logarithms).

The formula is particularly handy when you don't have a calculator that directly computes logarithms for bases other than 10 or \( e \). The formula is expressed as:
\[\log_{b} a = \frac{\log_{10} a}{\log_{10} b}\]This converts \( \log_{b} a \) into a ratio of common logarithms, which are typically easier to calculate using a standard calculator. For example, to convert \( \log_{3} 8 \), you calculate the common logarithms of 8 and 3 and then divide the two, making the computation straightforward.

Decimal Approximation

Decimal approximation is a way to express the value of a logarithm as a decimal number, rounded to a specified number of places. This is useful in most practical situations where an exact fraction isn't necessary or possible.

After converting a logarithm to a base that can be computed with a calculator, such as in our example \( \log_{3} 8 = \frac{\log_{10} 8}{\log_{10} 3} \), you perform the division to get a numerical result. To a suitable precision, this result is usually rounded. For our example, you find:

• \( \log_{10} 8 \approx 0.9031 \)
• \( \log_{10} 3 \approx 0.4771 \)

Then, the decimal approximation becomes:
\[ \log_{3} 8 \approx \frac{0.9031}{0.4771} \approx 1.8928 \]
This rounded result gives a practical number to work with, especially useful in fields where precision to many decimal places is often unnecessary.