Problem 26 - College Algebra with Corequisite Support

Question

For the following exercises, rewrite each expression as an equivalent ratio of logs using the indicated base. \(\log _{14}(55.875)\) to base 10

Step-by-Step Solution

Answer

\(\log_{14}(55.875) = \frac{\log_{10}(55.875)}{\log_{10}(14)}\) using base 10 logs.

Step 1: Identify the Given Logarithm

We are given the logarithm \(\log_{14}(55.875)\) and asked to rewrite it in terms of a ratio of logarithms with base 10.

Step 2: Use the Change of Base Formula

The change of base formula for logarithms is \( \log_b(a) = \frac{\log_c(a)}{\log_c(b)} \), where \(c\) is the new base. Here, we want to express the log using base 10, so we set \(c = 10\).

Step 3: Rewrite Using Base 10

According to the change of base formula: \( \log_{14}(55.875) = \frac{\log_{10}(55.875)}{\log_{10}(14)} \). This expresses \(\log_{14}(55.875)\) as a ratio of base 10 logs.

Key Concepts

Logarithmic expressions

Logarithmic expressions form the foundation for understanding various phenomena, especially in scientific and mathematical contexts. Essentially, a logarithm answers the question: "To what power must the base be raised, to produce a given number?"

Think of it as the reverse operation of exponentiation. If you have an equation like \( b^x = a \), the logarithmic form would be \( x = \log_b(a) \). Here, "b" is the base, "a" is the number you're examining, and "x" is the exponent. This is why the expression \( \log_b(a) \) gives the power that b must be raised to, so that it equals a.

When working with logarithmic expressions, it's helpful to familiarize yourself with properties such as:

• Product Rule: \( \log_b(xy) = \log_b(x) + \log_b(y) \)
• Quotient Rule: \( \log_b\left(\frac{x}{y}\right) = \log_b(x) - \log_b(y) \)
• Power Rule: \( \log_b(x^n) = n \cdot \log_b(x) \)

These rules greatly simplify handling more complex expressions and are particularly useful when changing the base of a logarithm.

Base 10 logarithms

Base 10 logarithms are also known as common logarithms. They're widely used in real-world applications, including science and engineering, because they relate closely to metric scales that are also base 10, like pH levels or the Richter scale for earthquakes.

When you see \( \log(x) \) without a specified base, it's typically understood to be base 10. This simplifies notation and handling because you don’t have to specify the base. In your calculator or computer, the "log" button usually operates as a base 10 logarithm.

Using the properties of logarithms with base 10 offers a streamlined way of solving problems. Here are a few practical points:

• Base 10 logarithms can help convert multiplication into addition, simplifying arithmetic operations.
• They're helpful for expressing very large or small numbers in a manageable way (like expressing "thousands" or "millionths").
• Understanding base 10 logs are crucial in scientific notation and for many scientific calculations.

All this makes base 10 logarithms not just a mathematical curiosity but a very practical tool.

Ratio of logs

The concept of the ratio of logs arises frequently when multiplying or dividing numbers represented as logarithms. This concept is quite powerful in simplifying complex computations involving different logarithmic bases.

The change of base formula, \( \log_b(a) = \frac{\log_c(a)}{\log_c(b)} \), which transforms any logarithm into a ratio of logs based on a new base c, is central to this topic. This transformation is particularly significant when you are limited to using base 10 logs, as most calculators readily provide these.

When you see a ratio of logs, you're effectively converting the problem into this form using another base, often for convenience. For example:

• This conversion is crucial when solving exponential equations where directly calculating in one base isn't possible or practical.
• In cases where exact calculation might be required, such as in scientific data analysis, using a ratio of logs ensures accuracy by circumventing rounding errors inherent in manual computations.

The ability to manipulate logarithms in this way is a valuable skill in many mathematical domains.

Logarithmic conversion

Logarithmic conversion involves changing a log from one base to another base. It's an essential operation, especially when dealing with different technological or scientific tools that may not support arbitrary bases natively.

This conversion is executed using the change of base formula: \( \log_b(a) = \frac{\log_c(a)}{\log_c(b)} \). By applying this formula, you can express a logarithm in whatever base is most convenient or required for your purposes. For instance:

If you need to calculate \( \log_2(16) \) but only have a tool that uses base 10 logs, convert it by using the formula:

• Set the new base \( c \) (often 10) to work within calculator restrictions.
• Apply: \( \log_2(16) = \frac{\log_{10}(16)}{\log_{10}(2)} \), simplifying the calculations.

Learning how to switch logarithmic bases with the conversion formula allows for flexibility and convenience in both theoretical problems and real-world applications. It ensures you're not restricted by the original base of a logarithm and can effectively solve logarithmic problems using whatever tools are available to you.