Problem 72 - College Algebra with Modeling and Visualization

Question

Use the change of base formula to approximate the logarithm to the nearest thousandth. $$ \log _{9} 85+\log _{7} 17 $$

Step-by-Step Solution

Answer

The approximate value is 3.477.

Step 1: Apply Change of Base Formula

The change of base formula states that \( \log_b a = \frac{\log_c a}{\log_c b} \), where \( c \) can be any positive number (commonly 10 or \( e \)). We'll use base 10 to apply the formula to each logarithm.

Step 2: Convert \( \log_9 85 \) Using Base 10

According to the change of base formula, \( \log_{9} 85 = \frac{\log_{10} 85}{\log_{10} 9} \). Calculate the values using a calculator: - \( \log_{10} 85 \approx 1.9294 \)- \( \log_{10} 9 \approx 0.9542 \)Thus, \( \log_{9} 85 = \frac{1.9294}{0.9542} \approx 2.021 \).

Step 3: Convert \( \log_7 17 \) Using Base 10

Similarly, apply the change of base formula to convert \( \log_{7} 17 \): \( \log_{7} 17 = \frac{\log_{10} 17}{\log_{10} 7} \). Using a calculator:- \( \log_{10} 17 \approx 1.2304 \)- \( \log_{10} 7 \approx 0.8451 \)Thus, \( \log_{7} 17 = \frac{1.2304}{0.8451} \approx 1.456 \).

Step 4: Add the Converted Logarithms

Now that we have both values:\( \log_{9} 85 \approx 2.021 \) and \( \log_{7} 17 \approx 1.456 \).Add these values to find the final answer: \( 2.021 + 1.456 = 3.477 \).

Key Concepts

Understanding Logarithms

Logarithms represent the power to which a given base must be raised to obtain a specific value. For example, if we have a logarithm like \( \log_b(a) \), it answers the question: "To what power must \( b \) be raised to yield \( a \)?"
An essential concept in logarithms is that they are the inverse operations of exponentiation. If you think of \( b^x = a \), then \( \log_b(a) = x \).

Another handy property of logarithms is that they convert multiplication into addition. This is particularly useful in solving complex exponential equations or simplifying calculations.
Logarithms can be used in different bases, with two of the most common being base 10 and the natural base \( e \). Understanding the basic concept of logarithms is foundational in mathematics and can be applied in various real-world contexts like computing interest rates or analyzing sound intensity.

Base 10 Logarithms Demystified

Base 10 logarithms, also known as common logarithms, are those where the base is 10. It's often written as simply \( \log(a) \) rather than \( \log_{10}(a) \).
Why base 10? It's because our number system is decimal-based, making it easy for calculations, especially with a calculator, where \( \log \) usually implies base 10.

To calculate a logarithm in base 10, think about your number 10^something to reach your original number. For example, \( \log(100) = 2 \) because 10 raised to the power of 2 gives you 100.

• Common logarithms are used in sciences to measure quantities that range over many orders of magnitude, such as pH in chemistry or Richter scale in geology.
• They simplify multiplication and division into addition and subtraction, making complex calculations more manageable.

Using tools like the change of base formula, we can convert other logarithmic bases into base 10, which simplifies the process of dealing with complicated bases.

Adding Logarithms Simplified

Adding logarithms can be a straightforward task when certain conditions are met. One of the main properties is that if two logarithms have the same base, you can combine them.
For instance, \( \log_b(M) + \log_b(N) = \log_b(M \cdot N) \). This shows that you can turn adding into multiplying which is extensively used in simplifying logarithmic expressions.

However, in cases where the bases are different, such as the exercise provided, you need to convert them first using the change of base formula.
Once both logarithms are in the same base, particularly base 10 for calculation ease, you can add their values straightforwardly.
Steps to add logarithms include:

• Convert all logarithms to a common base using the change of base formula.
• Perform the necessary calculations to find individual logarithm values.
• Add the results obtained to get the final result.

The understanding of adding logarithms is key in simplifying numerous math problems involving exponential growth and decay.