Problem 75 - College Algebra

Question

In Exercises \(71-78,\) use common logarithms or natural logarithms and a calculator to evaluate to four decimal places. $$ \log _{0.1} 17 $$

Step-by-Step Solution

Answer

The exact value of \(\log _{0.1} 17\) depends on the calculations but will follow the steps as described. It's approximated to four decimal places.

Step 1: Apply Change of Base Formula

In order to compute the logarithm in a different base, the change of base formula needs to be applied. This is expressed as: \[\log_b a = \frac{\log_k a}{\log_k b}\] where \(k\) can be any established base - either natural logarithm (\(e\)) or base \(10\). In this case, can use natural logarithm, denoted as \(\log\) or \(\ln\). Hence, the expression \(\log _{0.1} 17\) changes to \(\frac{\ln17}{\ln0.1}\).

Step 2: Compute the Logarithm Values

Use a calculator to find the values of \(\ln17\) and \(\ln0.1\).

Step 3: Calculate \(\log _{0.1} 17\)

Divide \(\ln17\) by \(\ln0.1\). This will give the value of \(\log _{0.1} 17\).

Step 4: Approximate the Value

Round off the value obtained from the previous step to four decimal places to reach the final answer.

Key Concepts

Change of Base Formula

When faced with a logarithmic expression that requires using a base other than base 10 (common logarithm) or base e (natural logarithm), the change of base formula comes to the rescue. This formula allows you to convert a logarithm from any base to a more familiar base, like 10 or e. The formula is given as follows:

• \[ \log_b a = \frac{\log_k a}{\log_k b} \]

Here, \(b\) is the base you wish to change from, and \(k\) is your chosen new base, generally either 10 or e. By utilizing this formula, you can easily compute logarithms with complicated bases using a calculator by converting them to a base you are comfortable working with.
For instance, if you wanted to compute \(\log_{0.1} 17\), you'd convert it to base e using the natural logarithm: \(\frac{\ln 17}{\ln 0.1}\). This way, you can simply input it into your calculator using familiar functions.

Natural Logarithms

Natural logarithms use the base \(e\), where \(e\) is an irrational number approximately equal to 2.71828. The natural logarithm is denoted by \(\ln\), and can be extremely useful in both calculus and other areas of mathematics, especially when dealing with exponential growth and decay.
When you see \(\ln\) in a mathematical expression, it indicates a logarithm with base \(e\). One aspect that makes the natural logarithm special is its relationship with the exponential function. For any number \(x\), the equation \(\ln(e^x) = x\) holds true, making it uniquely straightforward in simplifying expressions that involve exponentials.
To employ the change of base formula using natural logarithms, simply take the \(\ln\) of both the base and the number in the expression, just like in our example: \(\frac{\ln 17}{\ln 0.1}\). This aids in simplifying complex calculations, especially when natural logs exhibit certain properties advantageous to solving equations.

Base 10 Logarithms

Base 10 logarithms, commonly referred to as "common logarithms," use 10 as their base and are denoted simply as \(\log\). These are particularly helpful because base 10 reflects our standard numerical system. Most scientific calculators have a dedicated button for \(\log\), making these logarithms convenient to calculate.

• Unlike natural logarithms, there is no need to specify the base when working with base 10 because it is default when no base is indicated.
• For example, \(\log 100\) translates directly to the question, "10 raised to what power gives 100?" The answer here would be 2, because \(10^2 = 100\).

When using the change of base formula, base 10 logarithms offer a simple and practical way to evaluate logarithms of unconventional bases: simply replace both \(\log_k a\) and \(\log_k b\) with \(\log a\) and \(\log b\) respectively. Although in this specific exercise we used natural logarithms, using common logarithms would have worked just as effectively to find \(\log_{0.1} 17\), by calculating \(\frac{\log 17}{\log 0.1}\).
Therefore, understanding both natural and common logarithms enhances flexibility when tackling logarithmic problems.