Problem 55 - Algebra and Trigonometry

Question

Use the Change of Base Formula and a calculator to evaluate the logarithm, correct to six decimal places. Use either natural or common logarithms. $$ \log _{4} 125 $$

Step-by-Step Solution

Answer

\( \log_{4} 125 \approx 3.483229 \)

Step 1: Recall the Change of Base Formula

The Change of Base Formula allows us to evaluate logarithms with bases other than 10 or e. The formula is \( \log_b a = \frac{\log_c a}{\log_c b} \), where \( c \) is a number of our choice, commonly 10 (common logarithm) or e (natural logarithm).

Step 2: Apply the Change of Base Formula

Choose \( c = 10 \) (common logarithm) to apply the formula: \( \log_{4} 125 = \frac{\log_{10} 125}{\log_{10} 4} \).

Step 3: Calculate the logarithm of 125 using a calculator

Use a calculator to find \( \log_{10} 125 \). The result is approximately 2.096910.

Step 4: Calculate the logarithm of 4 using a calculator

Use a calculator to find \( \log_{10} 4 \). The result is approximately 0.602060.

Step 5: Divide the results

Use the results from Steps 3 and 4: \( \log_{4} 125 = \frac{2.096910}{0.602060} \). Perform the division to find the result.

Step 6: Round the answer to six decimal places

After dividing, the result is approximately 3.483229, already rounded to six decimal places.

Key Concepts

change of base formula

The Change of Base Formula is a handy tool when dealing with logarithms that have bases other than 10 or \( e \). This formula simplifies the calculation of such logarithms by converting them into more familiar terms. It is expressed as:

• \( \log_b a = \frac{\log_c a}{\log_c b} \)

where \( b \) is the base you are changing from, and \( c \) is the base you are changing to, often 10 (common logarithm) or \( e \) (natural logarithm). This means you can calculate \( \log_4 125 \) by choosing a base \( c \).
If you choose the common logarithm, \( c = 10 \), you would express it as \( \frac{\log_{10} 125}{\log_{10} 4} \). This makes it easier to compute using a calculator, as most calculators have log functions based on base 10 or \( e \).
So, remember, the main purpose of the Change of Base Formula is to transform a potentially complex logarithmic expression into a simpler form that can be easily calculated.

common logarithms

Common logarithms are logarithms with base 10. They are represented as \( \log_{10} x \) or simply \( \log x \) when the base is not specified. This is because the common logarithm is one of the two logarithmic functions commonly found on scientific calculators, allowing for easy computation.
Common logarithms are particularly useful in a variety of applications, such as scientific measurements and calculations involving orders of magnitude. When using the Change of Base Formula, selecting 10 as the base is often a convenient choice. For instance, to calculate \( \log_{4} 125 \), you can calculate it as:

• \( \log_{4} 125 = \frac{\log_{10} 125}{\log_{10} 4} \)

Computing these values using a calculator gives approximately 2.096910 for \( \log_{10} 125 \) and 0.602060 for \( \log_{10} 4 \). These values are then divided to find \( \log_{4} 125 \) as roughly 3.483229 when rounded to six decimal places.
As you can see, using common logarithms simplifies the process while leveraging calculator functionalities.

natural logarithms

Natural logarithms, marked by the symbol \( \ln x \), are logarithms with base \( e \), where \( e \) is an irrational constant approximately equal to 2.71828. The natural logarithm is unique because it appears naturally in mathematics, especially in calculus and functions describing growth and decay, such as exponential growth models.
When employing the Change of Base Formula, you might choose a natural logarithm if your problem involves exponential functions. Here's how you could use it for the problem \( \log_4 125 \):

• Convert the expression using natural logarithms: \( \log_4 125 = \frac{\ln 125}{\ln 4} \)

Calculating these values using a calculator, you find \( \ln 125 \) and \( \ln 4 \), then divide the results similar to common logarithms.
While not as commonly used in arithmetic problems on calculators as common logarithms, natural logarithms are essential in higher mathematics and scientific contexts. They offer another tool to simplify complex problems and achieve the most accurate results possible in fields that rely on the constant \( e \).