Problem 26 - College Algebra and Calculus: An Applied Approach

Question

Evaluate the logarithm. Round your result to three decimal places.\(\log _{8} 3\)

Step-by-Step Solution

Answer

The value of \( \log _{8} 3 \) rounded to three decimal points is approximately 0.528.

Step 1: Understand and Apply the Change of Base Formula

To evaluate \( \log _{8} 3 \) on a calculator, the Change of Base Formula needs to be used. The formula is: \(log_b a = \frac{log_d a}{log_d b}\), where a is the argument, b is the base of the logarithm, and d can be whatever base of logarithm you wish to use. For this problem, a = 3, b = 8, and let's choose d to be 10 (a common logarithm). Apply the change of base formula: \(log_8 3\) = \(\frac{log_{10} 3}{log_{10} 8}\)

Step 2: Compute the common logarithms

Using a calculator to compute the common logarithms in the numerator and the denominator gives \(log_{10} 3 ≈ 0.477\) and \(log_{10} 8 ≈ 0.903\).

Step 3: Divide the results

The final step is to divide the results to get the value of the original logarithm in base 8. Therefore, \(\frac{0.477}{0.903} ≈ 0.528\) when rounded to three decimal places.

Key Concepts

Logarithm Evaluation

Understanding how to evaluate logarithms is a critical skill in mathematics. Let's demystify it with a simple example. When you come across a logarithm such as \( \log _{8} 3 \), you're basically looking for the answer to the question: 'To what power must I raise 8 to get 3?' Now, this isn't something that can be calculated easily in your head, especially with non-integer bases and results.

Fortunately, we can employ a useful mathematical tool known as the Change of Base Formula to evaluate such logarithms. This formula allows us to convert a logarithm with any base to a logarithm with a base that is more convenient, typically base 10 or base \(e\), because calculators can handle these 'common' and 'natural' logarithms directly. Using this approach, as in our solution, makes it possible to solve logarithms that otherwise would be challenging to compute.

Common Logarithms

Common logarithms have a base of 10 and are represented as \( \log _{10} x \) or simply \( \log x \). They're labeled 'common' because, historically, the base-10 number system has been widely used in scientific and engineering work. The common logarithm of a number tells us how many times we must multiply 10 to obtain that number.

For instance, the common logarithm of 1000 is 3, since \( 10 \times 10 \times 10 = 1000 \). In the context of our example, we find the common logarithms of both 3 and 8 using a calculator because these are not powers of 10 that we can easily recognize. This is a typical scenario where the Change of Base Formula becomes particularly handy for evaluation.

Logarithmic Computation

After understanding common logarithms, logarithmic computation involves simplifying or solving logarithmic expressions using various methods. One crux of this process, as seen in our step by step solution, involves translating a logarithm to a more 'calculator-friendly' form. Computing the common logarithms is an intermediate step that provides values we can handle. Then, it's all about arithmetic—dividing the resulted common logarithms to find the value of the original expression.

It's important to note the precision of the final answer, as logarithmic computations on calculators are approximate. In the given example, we rounded our result to three decimal places to match the requested precision. This reflects the practical usage of logarithms, where exact values are often less critical than understanding the relationship between the numbers involved.