Problem 29 - Intermediate Algebra

Question

For Problems \(23-32\), approximate each of the following logarithms to three decimal places. $$ \log _{9} 0.0017 $$

Step-by-Step Solution

Answer

\( \log_{9} 0.0017 \approx -2.903 \).

Step 1: Convert Logarithm to Base 10

Since the given logarithm, \( \log_{9} 0.0017 \), is not in the common base 10 or base \( e \), we'll use the change of base formula: \( \log_{b} a = \frac{\log_{10} a}{\log_{10} b} \).

Step 2: Apply Change of Base Formula

Substitute \( a = 0.0017 \) and \( b = 9 \) into the change of base formula: \( \log_{9} 0.0017 = \frac{\log_{10} 0.0017}{\log_{10} 9} \).

Step 3: Calculate \( \log_{10} 0.0017 \)

Use a calculator to find \( \log_{10} 0.0017 \), which equals approximately \( -2.770 \).

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

Find \( \log_{10} 9 \) using a calculator, which is approximately \( 0.954 \).

Step 5: Compute the Result

Divide \( -2.770 \) by \( 0.954 \) to find the value of \( \log_{9} 0.0017 \). This calculation gives approximately \( -2.903 \).

Step 6: Round to Three Decimal Places

The result \(-2.903\) is already rounded to three decimal places. Therefore, \( \log_{9} 0.0017 \approx -2.903 \).

Key Concepts

Change of Base Formula

The change of base formula is a key tool in simplifying and computing logarithms with bases other than 10 or the natural base, which are often more intuitive to work with. It’s a simple yet powerful formula that allows you to convert any logarithm to a different base, typically base 10 (known as the common logarithm) or base e (the natural logarithm). The formula is expressed as:\[\log_b a = \frac{\log_c a}{\log_c b}\]where:

• \(a\) is the argument of the logarithm you're trying to compute,
• \(b\) is the original base of the logarithm,
• \(c\) is the new base you want to convert to (most commonly base 10).

This conversion makes calculation easier—especially with a calculator—as most calculators natively support calculations in base 10 or base e. For our problem, using the change of base formula allows us to convert \(\log_9 0.0017\) into two common logarithms: \(\log_{10} 0.0017\) divided by \(\log_{10} 9\). This is particularly helpful when dealing with bases that are not common as it breaks the problem into manageable parts.

Common Logarithm

A common logarithm is a logarithm with base 10. These are among the simplest and most often used types of logarithms in mathematical calculations because of their compatibility with calculators and their prevalence in logarithmic tables. It’s denoted as \(\log_{10} a\) or sometimes simply \(\log a\) when the base is understood to be 10.Common logarithms have many applications:

• They are used in scientific calculators for quick calculations,
• They simplify expressions and equations by converting multiplication into addition,
• They are applied in real-world scenarios, such as calculating pH in chemistry and measuring sound intensity in decibels.

In the given exercise, transforming \(\log_9 0.0017\) to the common logarithm form using the change of base formula simplifies the calculation as it gives us \(\log_{10} 0.0017\) and \(\log_{10} 9\), both of which can directly be computed using a calculator.

Decimal Approximation

Decimal approximation is a method used to express numbers that result from calculations such as logarithms to a desired precision, often resulting in a more practical, readable form. When you calculate logarithms, especially non-integer ones, they frequently result in long, sometimes unwieldy decimal numbers. To make these numbers useful for further computation or understanding, they are rounded to a specific number of decimal places. Here are some common practices:

• Deciding how many decimal places are necessary depends on the problem requirements,
• Logarithms are frequently approximated to three decimal places in educational settings,
• This rounding can aid in maintaining consistency in calculations.

In this exercise, calculating \(\log_{10} 0.0017\) gave us \(-2.770\) and for \(\log_{10} 9\) we obtained \(0.954\). After dividing these values, the resulting \(-2.903\) was already exactly to three decimal places, ensuring clarity and precision in the final answer \(\log_9 0.0017 \approx -2.903\).