Problem 70 - College Algebra with Modeling and Visualization

Question

Use the change of base formula to approximate the logarithm to the nearest thousandth. $$ \log _{6} 0.77 $$

Step-by-Step Solution

Answer

\( \log_6 0.77 \approx -0.146 \).

Step 1: Identify the Change of Base Formula

The change of base formula for logarithms is used to convert logarithms to a different base, typically base 10 or base e (natural logarithm). It is given by: \[ \log_b a = \frac{\log_c a}{\log_c b} \] where \( c \) is the new base of the logarithm, often 10 or \( e \).

Step 2: Apply the Change of Base Formula

Use the change of base formula to rewrite \( \log_6 0.77 \) using base 10. The expression becomes: \[ \log_6 0.77 = \frac{\log_{10} 0.77}{\log_{10} 6} \].

Step 3: Calculate \( \log_{10} 0.77 \) and \( \log_{10} 6 \)

Using a calculator, compute the common logarithms: \( \log_{10} 0.77 \approx -0.113943 \) and \( \log_{10} 6 \approx 0.778151 \).

Step 4: Divide the Logarithms

Substitute the computed values into the formula: \[ \log_6 0.77 = \frac{-0.113943}{0.778151} \approx -0.1464 \].

Step 5: Round to the Nearest Thousandth

Round the decimal result to the nearest thousandth: \( -0.1464 \) rounds to \( -0.146 \).

Key Concepts

Logarithms

Logarithms are a mathematical concept that allows us to express how many times we need to multiply a number (the base) to get another number. If you think of them, they are basically the opposite of exponentiation. For example, if 2 to the power of 3 equals 8, the logarithm base 2 of 8 is 3, or \( \log_2 8 = 3 \).
Logarithms are very helpful in dealing with exponential growth and decay, like that found in populations or radioactive materials. They transform complex multiplication into simpler addition, which can make calculations easier. Here’s why learning about logarithms is valuable:

• They simplify calculations involving large numbers.
• They help to solve equations involving exponential growth or decay.
• They provide a better understanding of phenomena that change rapidly or slowly over time.

Base Conversion for Logarithms

Base conversion for logarithms is a useful process when you need to change the base of a logarithm to another base, often to a simpler or more standardized one. The base conversion is usually done with the help of the Change of Base Formula, which provides a method to calculate logarithms that are not as easy to compute directly.
The formula is given by \( \log_b a = \frac{\log_c a}{\log_c b} \). In this formula, - \( b \) is your original base, - \( c \) is the new base you want to convert to, usually 10 (common logarithms) or \( e \) (natural logarithms), - and \( a \) is the number you are taking the logarithm of.
By converting to a new base, usually base 10, you can utilize calculators or computational software that often use this standardized base. This conversion becomes essential in fields like computer science and engineering, where calculations are frequent and need to be efficient and accurate.

Common Logarithms

Common logarithms are logarithms that use base 10. They are frequently used in mathematics and everyday life because our number system is base 10, making them simple and convenient for computations. When no base is mentioned in a logarithm notation, it is typically assumed to be 10, written simply as \( \log a \).
Using common logarithms has its advantages:

• It aligns with the decimal system which we use universally, simplifying many calculations.
• It is the default base in many calculators and computation tools, making it a practical choice when performing calculations.
• They allow for the easy computation of the logarithm of numbers without advanced scientific calculators.

Understanding how to use common logarithms effectively can greatly assist in solving various mathematical problems, especially those involving growth patterns or decay functions. They also facilitate easier communication of mathematical ideas, since they are widely recognized and used.