Problem 59 - Algebra and Trigonometry

Question

\(55-62\) . Use the Change of Base Formula and a calculator to evaluate the logarithm, rounded to six decimal places. Use either natural or common logarithms. $$ \log _{7} 2.61 $$

Step-by-Step Solution

Answer

\( \log_7 2.61 \approx 0.493227 \)

Step 1: Identify the Change of Base Formula

The Change of Base Formula allows us to evaluate logarithms with any base using known logarithms like common (base 10) or natural (base e) logarithms. The formula is given by: \[ \log_b a = \frac{ \log_c a }{ \log_c b } \]where \( b \) is the base of the original logarithm, \( a \) is the argument of the logarithm, and \( c \) is any new base, commonly 10 or \( e \).

Step 2: Apply the Change of Base Formula

To solve \( \log_7 2.61 \) using base 10 logarithms:\[ \log_7 2.61 = \frac{ \log_{10} 2.61 }{ \log_{10} 7 } \]

Step 3: Use a Calculator to Evaluate Logarithms

Using a calculator, find \( \log_{10} 2.61 \) and \( \log_{10} 7 \). You should get:- \( \log_{10} 2.61 \approx 0.416801 \)- \( \log_{10} 7 \approx 0.845098 \)

Step 4: Compute the Result

Substitute the values obtained into the Change of Base formula:\[ \log_7 2.61 = \frac{0.416801}{0.845098} \]Now, divide the found values: \( \frac{0.416801}{0.845098} \approx 0.493227 \).

Key Concepts

Understanding Logarithms

Logarithms are a fundamental mathematical concept used to determine the power to which a base number must be raised to obtain another number. For instance, if you know that \( b^x = a \), the logarithm helps you find \( x \). This concept was created to transform multiplicative relationships into additive ones, simplifying complex calculations.

A logarithm can be expressed as \( \log_b a \), where \( b \) is the base, and \( a \) is the argument. In practical scenarios, most commonly used logarithms are base 10 (common logarithm) and base \( e \) (natural logarithm).:

• Base 10 logarithms are typically written as \( \log_{10} \) or simply \( \log \).
• Natural logarithms use base \( e \), represented as \( \ln \).

Logarithms have several applications, from solving exponential growth problems to making scientific calculators easier to use.

Base Conversion Using Logarithms

Sometimes, a logarithm is given in a base that is not readily available on typical calculators, such as base 7 in the original exercise. To resolve this, you can use the Change of Base Formula, which enables conversion into either base 10 or natural base logarithms. This formula is:\[\log_b a = \frac{ \log_c a }{ \log_c b }\]This means you can compute a logarithm of any base using a more commonly used base.

The Change of Base Formula is applied in the original solution, where the base 7 logarithm \( \log_7 2.61 \) is converted to base 10:

• Compute \( \log_{10} 2.61 \).
• Compute \( \log_{10} 7 \).
• Divide the two results for the final answer.

This method is not just a great practice for manual calculations, but also a useful skill for efficiently utilizing calculators.

Calculator Usage for Logarithms

Most scientific calculators are equipped to compute logarithms for base 10 and base \( e \). For other bases, you have to rely on the Change of Base Formula. Let's go through using a calculator for this purpose:

• First, find the "log" button for base 10 calculations, or "ln" for natural logs.
• Input the number you need to calculate the logarithm for, such as 2.61 and 7 in the original solution.
• Record the outputs. In our solution, this was approximately 0.416801 for \( \log_{10} 2.61 \) and 0.845098 for \( \log_{10} 7 \).
• Finally, divide the results according to the Change of Base Formula.

In our exercise, using these steps allows you to find \( \log_7 2.61 \approx 0.493227 \). Utilizing your calculator effectively for logarithmic conversions is an essential skill, making complex math problems manageable.