Problem 14 - Algebra and Trigonometry Real Mathematics, Real People

Question

Evaluate the logarithm using the change-of-base formula. Round your result to three decimal places.$$\log _{9} 4$$.

Step-by-Step Solution

Answer

The resulting value of \( \log_{9} 4 \) rounded to three decimal places is 0.631.

Step 1: Recall the change-of-base formula

The change-of-base formula is: \( \log_a b = \frac{\log_c b}{\log_c a} \), where \(c\) can be any positive number different than 1.

Step 2: Substitute into the formula

Substitute \(a = 9\), \(b = 4\) and \(c = 10\) (base 10) into the formula: \( \log_9 4 = \frac{\log_{10} 4}{\log_{10} 9} \).

Step 3: Perform the calculations

Calculate the values making use of a calculator to accurately find the logarithm values. The calculation is: \( \frac{\log_{10} 4}{\log_{10} 9} = \frac{0.6021}{0.9542} \).

Step 4: Round to three decimal places

Following the instruction to round the final value to three decimal places, the solution is rounded to 0.631.

Key Concepts

Logarithms

Logarithms are a fascinating mathematical concept that can initially seem tricky but are really just another way of thinking about exponents. If you know about exponents, you've got a great start! In simple terms, a logarithm answers the question: Given a base, to what power must it be raised to produce a given number?
For example, with the expression \( \log_9 4 \), we are trying to determine what power you need to raise 9 to in order to get 4. This can be daunting, especially when you can't solve it by just guessing. That's where the change-of-base formula steps in, providing a way to solve these tricky problems by converting them to a base you might find more familiar, like base 10, otherwise known as the common logarithm.
It's crucial to understand how flexible logarithms can be. With the change-of-base formula, given by \( \log_a b = \frac{\log_c b}{\log_c a} \), you are using a tool that makes these calculations accessible even if they seem a bit complex at first sight.

Base 10

When we talk about base 10, we are discussing the common logarithm. It's the type of logarithm most frequently used, especially in calculators, making it practical when performing calculations. Base 10 logarithms are denoted as \( \log_{10} x \) or simply \( \log x \).
In the context of different bases, base 10 is quite special. Because of its widespread use in computational tools, it serves as a familiar ground for converting more complex expressions—like when utilizing the change-of-base formula. This is why, in our exercise, we chose \( c = 10 \) to help evaluate \( \log_9 4 \).
Base 10 logarithms make things simpler thanks to the readily available technology. Many scientific calculators are set to perform calculations using logarithms in base 10, which ensures convenience and uniformity in mathematical operations.

Rounding

Rounding is an essential concept in mathematics because it simplifies numbers to make them easier to work with or to express results in a clear, concise manner. This becomes particularly important when you deal with numbers that have many decimal places. In real-world applications, we often don't need an exact answer to multiple decimal places, so rounding gives us a practical solution.
When rounding numbers, you look at the digit immediately after the decimal place to which you want to round. In our solution of \( \log_9 4 \), we calculated the logarithm result as \( 0.631 \) rounded to three decimal places. Rounding to three decimal places helps achieve precision while keeping the number easy to read and use.
Remember, rounding decisions can impact the precision of your answer. Follow the usual rounding rules: rounding up if the next digit is 5 or greater, and keeping it the same if it is less than 5, to make sure your result is both accurate and practical.