Problem 69 - Algebra 2

Question

Express each logarithm in terms of common logarithms. Then approximate its value to four decimal places. \(\log _{50} 23\)

Step-by-Step Solution

Answer

\( \log_{50} 23 \approx 0.8016 \).

Step 1: Change of Base Formula

To express the logarithm \( \log_{50} 23 \) in terms of common logarithms, we use the change of base formula. The change of base formula states that \( \log_{b} a = \frac{\log_{c} a}{\log_{c} b} \). Often, the common base \( c \) used is 10, leading to common logarithms \( \log_{10} \), which is often written simply as \( \log \). Therefore, we can express \( \log_{50} 23 \) as \( \log_{50} 23 = \frac{\log 23}{\log 50} \).

Step 2: Calculate Common Logarithms

Next, we calculate the common logarithms \( \log 23 \) and \( \log 50 \) using a calculator. By calculation, we get: \( \log 23 \approx 1.3617 \) and \( \log 50 \approx 1.6990 \).

Step 3: Apply the Change of Base Formula

Substitute the calculated values into the change of base formula: \( \log_{50} 23 = \frac{\log 23}{\log 50} = \frac{1.3617}{1.6990} \).

Step 4: Compute the Result

Finally, divide the values: \( \frac{1.3617}{1.6990} \approx 0.8016 \). Therefore, \( \log_{50} 23 \approx 0.8016 \) when expressed in terms of common logarithms and calculated to four decimal places.

Key Concepts

Change of Base Formula

The change of base formula is a powerful tool in logarithmic calculations used to express a logarithm in terms of another base. This comes in handy when your calculator doesn't support logarithms of arbitrary bases but does support common logarithms. The formula is simple:

• For any positive numbers \(a\), \(b\), and a valid base \(c\), the logarithm \(\log_{b} a\) can be converted to another base as: \(\log_{b} a = \frac{\log_{c} a}{\log_{c} b}\).
• In many cases, the chosen base \(c\) is 10, which simplifies the calculation using common logarithms.

The conversion helps simplify complex logarithmic expressions by using more familiar calculations. It also allows the approximation of logarithmic values when calculators or software may not offer direct support for the original base.
In our exercise, we wanted to express \(\log_{50} 23\) using common logarithms. By applying the change of base formula, we convert it to \(\frac{\log 23}{\log 50}\) which is much simpler to compute.

Common Logarithms

Common logarithms are logarithms with base 10. They are denoted simply as \(\log\). Since the number 10 is intuitive and widely used, especially in scientific and engineering contexts, common logarithms simplify calculations. Here are some key points:

• Common logarithms eliminate the base from notation since \(\log 10 = 1\).
• They are readily available on most scientific calculators, enabling easy and quick computation.

In practical terms, when you see \(\log x\), it generally means \(\log_{10} x\).
This simplicity makes common logarithms very practical for hand calculations and rough estimates.
In our example, common logarithms were used to find \(\log 23\) and \(\log 50\), essential steps in changing the base of our original logarithmic expression.

Logarithmic Calculation

Logarithmic calculation involves using logarithmic identities and properties to solve problems. It's a skill critical in mathematics, enabling you to handle a wide range of equations and functions. Here's how it typically unfolds:

• First, identify the type of logarithm and its base. Often, using the common base 10 streamlines the process.
• Apply known formulas—like the change of base formula—to simplify calculations.
• Use tools such as calculators for precise computation of common logarithms.

For example, in our exercise, the calculations started with using a calculator to find \(\log 23\) and \(\log 50\).
These values were then used in the formula \(\frac{\log 23}{\log 50}\) to approximate the result of \(\log_{50} 23\) as \(0.8016\).
Accuracy to four decimal places was ensured using proper logarithmic calculation techniques.