Problem 75 - College Algebra Essentials

Question

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 value of \(\log_{0.1} 17\) to four decimal places is -1.2304

Step 1: Apply the change of base formula

We can apply the change of base formula to convert the base 0.1 logarithm into base 10 logarithm. This is done like so: \(\log_{0.1} 17 = \frac{log_{10} 17}{log_{10} 0.1}\)

Step 2: Compute the common logarithms

Now, we can approximate the common logarithms using a calculator. \(\log_{10} 17 \approx 1.2304\) and \(\log_{10} 0.1 = -1\). These values can be input into the numerator and denominator respectively.

Step 3: Complete the Calculation

Now we can simply divide the two values found in step 2: \(\frac{1.2304}{-1} = -1.2304\).

Key Concepts

Change of base formula

When dealing with logarithms, sometimes the base we're working with is not very convenient for manual calculation. This is where the change of base formula comes in handy. It allows us to convert any logarithm into a more manageable form. The formula is expressed as:
\[\log_b(a) = \frac{\log_c(a)}{\log_c(b)}\]where \(b\) is the original base, \(a\) is the number you're taking the log of, and \(c\) is the new base you'd like to convert to. Usually, we convert to base 10 or base \(e\) (natural logarithms).
Using this formula simplifies calculations, especially when using a standard calculator that only has settings for common and natural logarithms. It ensures accuracy and makes the problem-solving process more efficient. This was the cornerstone for solving our original exercise.

Common logarithms

Common logarithms are logarithms with base 10, also known as \(\log_{10}\). These are widely used when performing calculations manually or with basic scientific calculators. Base 10 is termed 'common' because it's the foundation of our number system and therefore often used in fields that involve base 10 operations, such as engineering and scientific work.
Another reason common logarithms are preferred is that they often provide a clear and precise window into exponential growth and decay problems. Engineers and scientists often use them to analyze phenomena like sound intensity and acidity, where calculations are simplified through base 10.
In our original exercise, \(\log_{10} 17\) approximated to 1.2304 was calculated using the fact that our calculators prefer base 10 operations.

Base 10 logarithm

Also known as the decadic logarithm, the base 10 logarithm involves using 10 as the base for log calculations. This means that for any number \(x\), \(\log_{10}(x)\) is the power to which 10 must be raised to obtain \(x\).
In our case, the problem required converting \(\log_{0.1} 17\) into base 10, which is a smoother operation for standard calculators. It yielded an easy-to-handle number: 1.2304 for \(\log_{10} 17\) and -1 for \(\log_{10} 0.1\). This makes dividing them straightforward.
Therefore, it underscores why the base 10 logarithm is often the go-to for logarithmic equations in real-world applications.

Natural logarithms

Natural logarithms, denoted as \(\ln\), use base \(e\), where \(e\) is an irrational constant approximately equal to 2.71828. Natural logarithms are particularly significant because they tend to simplify derivatives and integrals in calculus, making them a preferred choice in mathematical analysis.
Moreover, they occur naturally in growth processes and continuous compounding scenarios, such as interest calculations and population growth models. While not used in the original exercise, understanding natural logs helps in broader calculations and transformations that require in-depth mathematical insights.
In solving logarithms where base \(e\) would make the calculations simpler or more precise, using natural logarithms becomes invaluable. Knowing when to use each type of logarithm allows for flexibility in problem-solving and leverages the strengths of the available mathematical tools.