Problem 113 - Algebra and Trigonometry

Question

Are based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam. Use the Change-of-Base Formula and a calculator to evaluate \(\log _{7} 62\). Round the answer to three decimal places.

Step-by-Step Solution

Answer

2.120

Step 1 - Recall the Change-of-Base Formula

The Change-of-Base Formula is used to evaluate logarithms with bases other than 10 or e. The formula is \(\text {log }_{b}(a) = \frac{\text{log }(a)}{\text{log }(b)}\). This formula converts the logarithm to a base that can be calculated using a standard calculator.

Step 2 - Apply the Change-of-Base Formula

Apply the Change-of-Base Formula to \(\text{log }_{7}(62)\). This becomes \(\text{log }_{7}(62) = \frac{\text{log }(62)}{\text{log }(7)}\).

Step 3 - Calculate Using a Calculator

Use a calculator to find \(\text{log }(62)\) and \(\text{log }(7)\). First, \(\text{log }(62) \) is approximately 1.792. Next, \(\text{log }(7)\) is approximately 0.845. Therefore, the expression becomes \( \frac{1.792}{0.845}\).

Step 4 - Perform the Division

Divide the values obtained: \(\frac{1.792}{0.845} ≈ 2.120\). This is the value of \(\text{log }_{7}(62)\) rounded to three decimal places.

Key Concepts

logarithms

Logarithms might sound complicated, but they're a powerful mathematical tool that helps us solve equations involving exponential functions. When we talk about logarithms, we often use the term 'log.' This is essentially the inverse operation of exponentiation. For instance, if we have an equation like \(b^y = x \), the logarithm of x to the base b is represented as \(log_b(x) = y\). In simpler terms, the logarithm tells us the power to which the base must be raised to get the number x.
There are two special types of logarithms commonly used:

• Common logarithms (base 10): Represented as \(log(x)\), which is the same as \(log_{10}(x)\).
• Natural logarithms (base e): Represented as \(ln(x)\), which is the same as \(log_e(x)\).

Understanding logarithms will make it easier to grasp other mathematical concepts, especially those involving exponential growth, decay, and finance.

calculator usage

Using a calculator can significantly simplify complex mathematical operations, including evaluating logarithms. When you need to calculate a logarithm with a base other than 10 or e, you'll use the Change-of-Base Formula. This formula restates the logarithm in terms of common or natural logarithms, which are easily accessible on most calculators.
To use the formula \(log_b(a) = \frac{log(a)}{log(b)}\):

• First, input the value inside the logarithm (a) and take its common (or natural) logarithm.
• Second, input the base of the logarithm (b) and take its common (or natural) logarithm.
• Finally, divide the first result by the second result to find the value of \(log_b(a)\).

Most scientific calculators have buttons for \(log\) and \(ln\) right on the keypad, making it very straightforward to perform these calculations.

base conversion

Base conversion is a useful technique for rewriting a number in one base to another base. This is crucial when dealing with logarithms that have bases other than 10 or e since most calculators only directly support these common bases.
The Change-of-Base Formula \(log_b(a) = \frac{log(a)}{log(b)}\) allows you to convert any logarithm to a common base that your calculator can handle. Here’s how to visualize this process:

• You start with a logarithm that has a different base, like \(log_7(62)\).
• Using the Change-of-Base Formula, you convert this logarithm to base 10, becoming \(\frac{log(62)}{log(7) }\).
• Both \(log(62)\) and \(log(7)\) are easy to input into your calculator.
• You then divide the results to get the final value.

Base conversion makes evaluating logarithms manageable and is essential for simplifying these kinds of mathematical operations.