Problem 90 - A Graphical Approach to College Algebra

Question

Use the change-of-base rule to find an approximation for each logarithm. $$\log _{9} 12$$

Step-by-Step Solution

Answer

\(\log_9 12 \approx 1.1309\) using the change-of-base formula.

Step 1: Recall the Change-of-Base Formula

The change-of-base formula for logarithms is used to rewrite a logarithm with a different base. It states that \(\log_b a = \frac{\log_c a}{\log_c b}\), where \(c\) is a new base that is usually 10 or \(e\) (common or natural logarithm).

Step 2: Apply the Change-of-Base Formula

Let's apply the change-of-base formula to \(\log_9 12\) using base 10:\[\log_9 12 = \frac{\log_{10} 12}{\log_{10} 9}\]

Step 3: Calculate \(\log_{10} 12\) and \(\log_{10} 9\)

Use a calculator to find the logarithms: \(\log_{10} 12 \approx 1.07918\) and \(\log_{10} 9 \approx 0.95424\).

Step 4: Divide the Logarithms

Now, divide \(\log_{10} 12\) by \(\log_{10} 9\):\[\frac{1.07918}{0.95424} \approx 1.1309\]

Step 5: Present the Approximation

The approximate value of \(\log_9 12\) using the change-of-base formula is 1.1309.

Key Concepts

Logarithms

Logarithms are a fundamental concept in mathematics, offering a way to explore exponential relationships in a different light. Simply put, a logarithm answers the question: "To what power must a specific base be raised to obtain a certain number?" This might sound complex initially, but with practice, it becomes more intuitive.

• Think of the base as the number that is repeatedly multiplied.
• The logarithm itself represents the number of times that multiplication occurs.

For example, if you have \(\log_2 8\), it means you're looking for how many times you need to multiply 2 to get 8. The answer is 3, because \(2^3 = 8\). This graphical or conceptual approach helps in understanding logarithms not just as numbers, but as counts of multiplication.
Logarithms have various applications, such as in calculating growth rates, data transformation for linear analysis, and in fields ranging from science to finance. They're particularly helpful in dealing with data that cover a large range of values.

Common Logarithm

A common logarithm, denoted as \(\log_{10} x\), is simply a logarithm with base 10. This type of logarithm is often used because we naturally think in base 10 due to our decimal number system.

• In mathematical terms, \(\log_{10} x\) is the power to which 10 must be raised to result in \(x\).
• Common logarithms are often used in scientific calculations, including the measurement of sound intensity (decibels), the pH level in chemistry, and in solving exponential decay problems.

While calculators typically provide a direct function for \(\log_{10} x\), common logarithms are particularly useful in the change-of-base formula for converting bases to a more familiar size. Most scientific fields prefer this due to its practicality in dealing with large and small numbers in computations, where powers of ten are a norm.

Natural Logarithm

The natural logarithm, represented as \(\ln x\), uses the mathematical constant \(e\) (approximately 2.71828) as its base. \(e\) is a unique number in mathematics, found throughout nature and calculus owing to its compound growth properties.

• The function \(\ln x\) gives the power to which \(e\) must be raised to get \(x\).
• Natural logarithms frequently appear in calculus, particularly in derivatives and integrals dealing with exponential growth and decay.

A distinguishing feature of \(\ln\) is its natural occurrence in math problems involving continuous growth or decay, such as population models and radioactive decay. In many ways, \(\ln\) provides more insight into advanced mathematics purely due to its connection to the rate of exponential change.