Problem 47 - Algebra 2

Question

REVIEW \(\log _{9} 30=\) $$ \begin{array}{l}{\mathbf{F} \log _{10} 9+\log _{10} 30} \\ {\mathbf{G} \log _{10} 9-\log _{10} 30} \\ {\mathbf{H}\left(\log _{10} 9\right)\left(\log _{10} 30\right)} \\ {\mathbf{J} \frac{\log _{10} 30}{\log _{10} 9}}\end{array} $$

Step-by-Step Solution

Answer

The correct answer is J.

Step 1: Understand the Problem

We need to evaluate \( \log_{9} 30 \) and choose the right option from the given choices. To do this, we should convert \( \log_{9} 30 \) using a change of base to base 10 or another easy base.

Step 2: Apply the Change of Base Formula

The change of base formula states that \( \log_{b} a = \frac{\log_{c} a}{\log_{c} b} \). For this problem, we'll use base 10: \( \log_{9} 30 = \frac{\log_{10} 30}{\log_{10} 9} \).

Step 3: Compare with the Options

The expression we derived from the change of base formula \( \frac{\log_{10} 30}{\log_{10} 9} \) matches option \( \mathbf{J} \).

Key Concepts

Change of Base Formula

The change of base formula is a powerful tool in logarithms. It helps to simplify the calculation of one base into another, such as base 10, which is commonly used in calculators. The formula states:
\[\log_b a = \frac{\log_c a}{\log_c b}\]
Here’s how it works:

• \(a\) is the value whose logarithm you are trying to find.
• \(b\) is the base of the logarithm you are changing from.
• \(c\) is the base you are changing to, usually base 10 for convenience in calculations.

To apply this to \(\log_{9} 30\), we substitute as follows:
\[\log_{9} 30 = \frac{\log_{10} 30}{\log_{10} 9}\]
This converts the logarithm from base 9 to base 10, making it easier to evaluate using a calculator. The formula is useful for comparing and calculating logarithms with any base flexibility.

Base 10 Logarithm

A base 10 logarithm, also known as a common logarithm, is widely used in mathematics and science. It's written as \(\log_{10}\) or simply as \(\log\) when the base is understood to be 10.
Why base 10?

• This base is linked to the decimal system. It provides a convenient way to express loudness in decibels or the pH in chemistry.
• Calculators are usually set to calculate base 10 logarithms. This means it’s easy to compute without needing extra adjustments or settings.

For example, when stating \(\log_{10} 100 = 2\), we mean that 10 raised to the power of 2 equals 100.
Understanding the common logarithm eases the process of solving real-world problems and applies to converting other bases using the change of base formula.

Logarithmic Functions

Logarithmic functions play a crucial role in understanding exponential decay, growth, and modelling. A logarithmic function is typically represented as \( f(x) = \log_b x \), where \(b\) is the base of the logarithm.
What makes them special?

• They are the inverse of exponential functions (e.g., \(b^y = x\) translates to \(y = \log_b x\)).
• They demonstrate a slower rate of increase compared to exponential functions, which is useful when studying growth patterns and trends.
• They are defined for positive real numbers, emphasizing their application in practical, real-world scenarios.

By utilizing properties of logarithms, such as product, quotient, and power rules, we can simplify complex equations and solve various types of mathematical problems effectively. These functions give insight into data spread and help in transformations essential for statistical analysis and graphical interpretations.