Problem 15 - Algebra and Trigonometry Real Mathematics, Real People

Question

Evaluate the logarithm using the change-of-base formula. Round your result to three decimal places.$$\log _{1 / 2} 16$$.

Step-by-Step Solution

Answer

Therefore, the approximate value of \(\log _{1 / 2} 16\) to three decimal places is -4.000.

Step 1: Apply Change-of-Base Formula

Start off by applying the change-of-base formula to the given expression. Due to its convenience, we'll use base 10 for the equivalent expression. Therefore, we rewrite the expression \(\log _{1 / 2} 16\) as \(\frac{\log 16}{\log 1/2}\).

Step 2: Calculate Logarithms

Next, use your calculator to find the common logarithms of 16 and 1/2. The common logarithm of 16, denoted \(\log 16\) is about 1.204. And the common logarithm of 1/2, denoted \(\log 1/2\), is about -0.301.

Step 3: Perform Division

Finally, divide the logarithm of 16 by the logarithm of 1/2, i.e., compute \(\frac{1.204}{-0.301}\). This yields a result of approximately -4.000 when rounded to three decimal places.

Key Concepts

Change-of-Base Formula

The change-of-base formula is a handy tool in logarithms that allows you to evaluate logarithms with any base. This formula states that if you have a logarithm of the form \(\log_b{a}\), you can rewrite it as \(\frac{\log_k{a}}{\log_k{b}}\), where \(k\) is a new base of your choice. Most often, people choose base 10 or the natural logarithm base \(e\), because they are easy to compute with calculators.
When dealing with unusual or more complicated bases, the change-of-base formula simplifies the process by using more conventional bases. This is especially useful when the calculator only supports common logarithms (base 10) or natural logarithms (base \(e\)). To put it simply, the change-of-base formula converts challenging logarithm problems into a format that's much easier to solve.

Common Logarithms

Common logarithms are logarithms with a base of 10, often denoted as \(\log(x)\) without explicitly writing out the base. That might sound intimidating at first, but it's actually quite simple! These types of logarithms were named 'common' because of their widespread use in science and engineering, where the decimal system is predominantly applied.
For many calculators, when you see the \(\log\) button, it's referring to the common logarithm with base 10. This can be very convenient when applying the change-of-base formula, as you only have to input \(\log{a}\) and \(\log{b}\) using readily available calculator functions.

• Easy to use with most calculators.
• Applications in scientific fields.
• Simplifies base conversions.

Whether you're dealing with chemistry or physics, recognizing that \(\log\) typically means base 10 can be a time-saver and make calculations less daunting.

Base 10

The base 10 system is fundamentally tied to our understanding of numbers, and it plays a critical role in logarithms. In mathematics, the base of a logarithm dictates the number being repeatedly multiplied. For base 10, it’s about powers of 10. For example, \(\log_{10}{100} = 2\) because 10 multiplied by itself twice equals 100.
This concept becomes crucial when you need to solve logarithmic problems where the simplicity of powers of 10 comes into play. The base 10 logarithm, or common logarithm, expresses how many times you multiply 10 to get the desired number.
Aside from ease in calculation, it also links closely to our numerical system—think of our banking, science, and even technological systems that largely depend on decimal structures.

• Easy conversion with powers of 10.
• Integral to the decimal numeric system.
• Supports comfortable calculation with practical applications.

Understanding the role of base 10 in logarithms helps create a bridge between abstract algorithms and tangible numerical processes.