Question

Describe two ways to evaluate \(\log _7\) 12 using a calculator.

Short answer

The two methods are: Direct calculation (if the calculator allows different bases) or the use of change of base formula (if the calculator does not allow different bases).

Step-by-step solution

Method 1: Direct Calculation

Some calculators have the capability to compute any base logarithm directly. Simply, type \(\log _7 12\) into your calculator and the result should appear. If your calculator allows you to change the base from the default base 10, ensure the base is set to 7, then just type in '12'.

Method 2: Change of Base Formula

If direct calculation is not possible, this method can be used. The change of base formula is defined as \(\log_b a = \frac{\log_k a}{\log_k b}\). So, to evaluate \(\log _7 12\), one can opt to use base 10 (on which most calculators operate), the calculation will then look like this: \(\log _7 12 = \frac{\log 12}{\log 7}\). This division can be computed on the calculator.

Key concepts

Change of Base Formula

The change of base formula is a handy tool when it comes to computing logarithms, especially when your calculator doesn't allow for direct base calculations. This formula transforms a logarithm in one base to a division equation in another base that your calculator understands, usually base 10 or base e (natural logarithm). It is defined as follows: \[\log_b a = \frac{\log_k a}{\log_k b}\]where \(b\) is the original base of the logarithm, \(a\) is the number whose logarithm you're trying to find, and \(k\) is the new base you are converting to. For the exercise given, we want to calculate \(\log_7 12\), so we can use the formula like so: \[\log_7 12 = \frac{\log_{10} 12}{\log_{10} 7}\] By evaluating \(\log 12\) and \(\log 7\) separately using the base 10 log function on your calculator, and then dividing the results, you can find the solution efficiently.

Logarithmic Functions

Logarithms are powerful mathematical tools that help us easily handle very large or very small numbers. A logarithm answers the question: "To what exponent must the base be raised in order to obtain a certain number?"

• The typical logarithmic function is written as \(\log_b a\), where \(b\) is the base and \(a\) is the number you are interested in.
• When the base, \(b\), is 10, it is often referred to as the common logarithm, and when the base is \(e\), a constant approximately 2.718, it is known as the natural logarithm.

Understanding how logarithmic functions work is essential in many fields like science, engineering, and finance. In our problem: \(\log_7 12\), you are trying to find the power to which 7 must be raised to produce 12.

Calculator Usage

Calculators have become essential for solving logarithmic problems, especially when they involve bases other than 10. Here's a quick guide on how to make the best use of your calculator when working with logarithms:

• Check if your calculator can change the base directly. Some scientific calculators allow you to enter custom bases using a function key, simplifying the process for bases other than 10.
• If your calculator can't handle custom bases, use the change of base formula. This involves using the common logarithm (base 10) or natural logarithm (base e) to perform your calculations.
• Verify your calculator's mode, ensuring it is set to calculate in degrees or radians if trigonometric functions are involved, although this is less common in basic logarithmic calculations.

For many problems, entering the two logarithms involved separately and performing a simple division is sufficient, just like in our example, \(\log_7 12 = \frac{\log 12}{\log 7}\). By mastering these calculator functions, you’ll find that logarithmic problems become much easier to manage.