Problem 61 - 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. $$ \log _{4} 125 $$

Step-by-Step Solution

Answer

\( \log_{4} 125 \approx 3.482203 \).

Step 1: Identify the Base and Argument

The problem requires you to find \( \log_{4} 125 \). Here, 4 is the base and 125 is the argument of the logarithm.

Step 2: Utilize the Change of Base Formula

The Change of Base Formula states that \( \log_{b} a = \frac{\log_{c} a}{\log_{c} b} \), where \( c \) is any positive number, commonly used bases are 10 (common logarithm) and \( e \) (natural logarithm).

Step 3: Calculate Using Common Logarithms

Let's use common logarithms where the base is 10. Hence, \( \log_{4} 125 = \frac{\log_{10} 125}{\log_{10} 4} \). Use a calculator to find \( \log_{10} 125 \approx 2.09691 \) and \( \log_{10} 4 \approx 0.60206 \).

Step 4: Compute the Final Expression

Substitute the calculated values into the formula: \( \log_{4} 125 = \frac{2.09691}{0.60206} \approx 3.4822026 \). Round it to six decimal places to get \( 3.482203 \).

Key Concepts

Logarithms

Logarithms are a fundamental concept in mathematics that help us deal with very big or very small numbers. A logarithm answers the question: **how many times do we multiply a certain number (the base) to get another number?** For example, if we want to find out how many times we need to multiply 4 to get 125, we write this using logarithms as \( \log_4 125 \). Here, 4 is the base, and 125 is the number we want to reach. Logarithms can be incredibly useful in simplifying difficult multiplications and handling exponential growth.

Common logarithms

Common logarithms are a type of logarithm where the base is 10. They are often used because our number system is based on 10. We write common logarithms as \( \log_{10} \) or simply \( \log \). For instance, suppose you want to calculate \( \log_{10} 1000 \). You ask how many times you need to multiply 10 to get 1000, which is 3, so: \( \log_{10} 1000 = 3 \). They are convenient for performing calculations with a calculator because they are usually pre-programmed to compute common logarithms.

Natural logarithms

Natural logarithms are logarithms with base \( e \), a special number approximately equal to 2.71828. We denote them as \( \ln \). They are crucial in many areas of science and mathematics because they relate closely to exponential functions. Much of calculus, especially the mathematics of growth and decay, involves \( e \) and natural logarithms. For example, if you need to solve for \( x \) in the equation \( e^x = 20 \), you use the natural logarithm to find that \( x = \ln(20) \). Natural logarithms make calculations with exponential growth more straightforward and are often encountered in continuous growth rate problems.

Base conversion

Base conversion is a technique used in logarithms to switch from one base to another. This is important because sometimes the base given isn't the one we can easily work with, whether for manual computing or using a calculator. The Change of Base Formula allows us to convert a logarithm to a new base. The formula is \( \log_b a = \frac{\log_c a}{\log_c b} \), where \( c \) can be any positive number. Commonly used bases for conversions are 10 and \( e \) since calculators can quickly handle these. For example, converting \( \log_{4} 125 \) using common logarithms involves calculating \( \frac{\log_{10} 125}{\log_{10} 4} \). This way, we can utilize the bases (10 or \( e \)) supported by technological tools like calculators.