Problem 18 - Algebra 2 and Trigonometry

Question

In \(15-23,\) evaluate each logarithm to the nearest hundredth. $$ \log 9+\log 3 $$

Step-by-Step Solution

Answer

\( \log 9 + \log 3 \approx 1.43 \)

Step 1: Apply the Product Property of Logarithms

Start by using the product property of logarithms, which states that \( \log_a b + \log_a c = \log_a (b \, c) \). In this case, \( \log 9 + \log 3 = \log (9 \times 3) \).

Step 2: Calculate the Product Inside the Logarithm

Next, calculate the product of 9 and 3. \[ 9 \times 3 = 27 \]So, \( \log (9 \times 3) = \log 27 \).

Step 3: Evaluate the Logarithm

Use a calculator to find \( \log_{10} 27 \) to the nearest hundredth.\[ \log 27 \approx 1.43 \]

Step 4: Round to the Desired Precision

Ensure that the calculated logarithm is rounded to the nearest hundredth, which we already did in the previous step:\[ \log_{10} 27 \approx 1.43 \]

Key Concepts

Product Property of Logarithms

The product property of logarithms is a fundamental rule that helps simplify expressions involving the sum of logarithms. This property states that the logarithm of a product is equal to the sum of the logarithms of its factors. It can be mathematically expressed as:

• \( \log_a b + \log_a c = \log_a (b \times c) \)

To see how this works, consider the expression \( \log 9 + \log 3 \). Using the product property, you can combine these two logarithms into one by multiplying the numbers inside:

• \( \log 9 + \log 3 = \log (9 \times 3) \)
• \( \log 27 \)

This transformation illustrates how the product property can simplify calculations, especially when evaluating sums of logarithms. It reduces the number of steps needed to reach an answer and can be particularly useful when dealing with complex logarithmic expressions.

Evaluating Logarithms

Evaluating logarithms involves finding the power to which the base must be raised to produce a given number. In most cases, we use calculators to evaluate logarithms, especially when they aren't simple integers. For instance, you might want to find the value of \( \log_{10} 27 \). This requires determining the power of 10 needed to result in 27.
To evaluate \( \log 27 \), input it into a calculator set to base 10 logarithms, since that's the default base. You'll get a decimal value.

• Using a calculator, \( \log 27 \approx 1.43 \)

Once you have this result, rounding it to the nearest hundredth gives a clear and precise answer. Evaluating logarithms accurately is crucial, as it allows for correct interpretations of data and reinforces understanding of logarithmic concepts in various applications.

Base 10 Logarithms

Base 10 logarithms, also known as common logarithms, use 10 as the base. This is the most frequently used logarithm, especially in scientific contexts due to its simplicity and the decimal number system. When we write \( \log 27 \), it typically implies a base 10 logarithm, \( \log_{10} 27 \).
Common logarithms have several practical applications, such as:

• Calculating growth rates
• Measuring sound intensity (decibel scale)
• Assisting in data analysis and normalization

The notation simplifies expressions since base 10 is both intuitive and widely recognized in various fields. In contexts where the base isn't specified, it's usually safe to assume it's base 10. Understanding and working with base 10 logarithms is vital for students, as it lays the foundation for more advanced logarithmic and mathematical concepts.