Problem 7 - Precalculus: Functions and Graphs

Question

Estimate using the change of base formula. $$\log _{9} 0.2$$

Step-by-Step Solution

Answer

Use the change of base formula: \(\log _{9} 0.2 \approx -0.7324\).

Step 1: Identify the Concept

The problem involves estimating the logarithm with a non-standard base. To solve it, we can use the change of base formula to rewrite the logarithm in terms of common logarithms like base 10 or base e (natural logarithm).

Step 2: Apply Change of Base Formula

The change of base formula states that \(\log_b a = \frac{\log_c a}{\log_c b}\). We will use base 10 logarithms to apply this to our problem: \(\log_9 0.2 = \frac{\log_{10} 0.2}{\log_{10} 9}\).

Step 3: Calculate \(\log_{10} 0.2\)

Using a calculator, find the common logarithm of 0.2. The result is \(\log_{10} 0.2 \approx -0.69897\).

Step 4: Calculate \(\log_{10} 9\)

Using a calculator, find the common logarithm of 9. The result is \(\log_{10} 9 \approx 0.95424\).

Step 5: Estimate the Logarithm

Now substitute the calculated values into the formula: \(\log_9 0.2 = \frac{-0.69897}{0.95424} \approx -0.7324\).

Step 6: Interpret the Result

The negative result indicates that 0.2 is less than 1, which is consistent with logarithms of numbers between 0 and 1.

Key Concepts

common logarithms

Common logarithms are a type of logarithm that use 10 as their base. They are often symbolized simply as \(\log\) when no base is written, as it is generally understood to be base 10. This makes common logarithms very important because they simplify many types of calculations.

• Base 10 fits naturally with our decimal numbering system, making common logarithms especially useful in various applications such as scientific and engineering calculations.
• Common logarithms are convenient when using calculators, as this base is typically the default setting for most devices.

Common logarithms can solve complex exponential problems and simplify the multiplication of large numbers by reducing them to addition or subtraction processes. Their consistent use in mathematical and practical real-world applications makes them a fundamental tool in mathematical problem-solving.

base 10 logarithms

Base 10 logarithms, also known as common logarithms, express a number as a power of 10. This means they show how many 10s you need to multiply together to get the original number. When working with a base 10 logarithm, we use the formula \(\log_{10} a\).

• For example, \(\log_{10} 100 = 2\) because 10 squared equals 100.
• These logarithms are especially relevant when dealing with data on a macroscopic scale, such as pH levels or decibel intensities, each frequently described using base 10 logarithms.

When comparing values of very different magnitudes, a base 10 logarithm can be crucial. They transform multiplicative processes into more manageable additive processes, which simplifies many types of calculations especially when predictability and ease of use are needed.

logarithms

Logarithms are the inverse of exponentiation, much like how division is the inverse of multiplication. If you know the product, using a logarithm can help you work backwards to find the original numbers. Logarithms have several key properties and uses:

• They transform multiplicative relationships into additive ones, which can make computations much simpler.
• Logarithms help to solve for unknowns in exponential equations by allowing us to "strip away" the exponents.
• They provide a method of working with very large or very small numbers on a more manageable scale, often useful in scientific disciplines.

Understanding logarithms is essential in fields ranging from computer science to acoustics, as they provide insights into how quantities grow or diminish exponentially. Familiarity with logarithms, including those with different bases, such as 10 or \(e\), broadens the scope of problems you can solve.

non-standard base

Logarithms are not restricted to base 10; they can have any positive base other than 1. When the base is neither 10 nor \(e\) (the base for natural logarithms), it is often referred to as a non-standard base.

• A common scenario involves changing these non-standard bases to a more familiar one like base 10, using the change of base formula: \(\log_b a = \frac{\log_c a}{\log_c b}\).
• This formula becomes particularly useful when you need to calculate a logarithm in a base that is not readily available on a standard calculator.

The change of base formula allows flexibility and consistency across different mathematical calculations. It empowers you to estimate or calculate values accurately regardless of the base involved. Understanding non-standard base logarithms and how to convert them using this formula is crucial for tackling a broader range of mathematical problems.