Problem 59 - Algebra A Combined Function

Question

Approximate each logarithm to four decimal places. $$ \log _{1 / 2} 5 $$

Step-by-Step Solution

Answer

\( \log _{1 / 2} 5 \approx -2.3219 \).

Step 1: Rewrite the Logarithm in terms of Natural Logarithms

We start by rewriting the given logarithm using the change of base formula. The change of base formula states that \( \log_{b} a = \frac{\log_c a}{\log_c b} \), where \( c \) can be any positive number (usually 10 or \( e \)). Here, we'll use the natural logarithm, \( \ln \), so the expression becomes: \( \log_{1/2} 5 = \frac{\ln 5}{\ln (1/2)} \).

Step 2: Calculate the Natural Logarithm Values

Now, calculate \( \ln 5 \) and \( \ln (1/2) \) using a calculator:- \( \ln 5 \approx 1.6094 \)- \( \ln (1/2) \approx -0.6931 \)

Step 3: Divide the Natural Logarithms

Next, divide the values obtained from the natural logarithms:\[\frac{\ln 5}{\ln (1/2)} = \frac{1.6094}{-0.6931} \approx -2.3219\]

Step 4: Approximate the Result to Four Decimal Places

Finally, approximate the result to four decimal places. The division yields \(-2.3219\), which is already in the required form.

Key Concepts

Natural Logarithms

Natural logarithms are logarithms with the base being the mathematical constant \( e \). The symbol \( \ln \) is used to denote a natural logarithm, such as \( \ln x \). The constant \( e \) is approximately 2.71828 and is an important number in mathematics, often arising in problems involving exponential growth or decay.

Natural logarithms help simplify certain calculations because they relate well to calculus due to the derivative properties of exponential functions. When we say \( \ln a \), we mean the power to which \( e \) must be raised in order to obtain \( a \). This property makes them very useful, especially in scientific and engineering contexts.

Change of Base Formula

The change of base formula is a powerful tool that allows us to rewrite logarithms in terms of a different base. The formula is:

• \( \log_{b} a = \frac{\log_{c} a}{\log_{c} b} \)

Here, \( c \) is the new base we are converting to, and \( c \) can be any positive number except 1. Often, we choose \( 10 \) or \( e \) because they are common bases for logarithms.

Using the change of base formula allows us to handle logarithms with exotic bases, facilitating easier computations. This is because calculators typically have buttons only for base 10 and base \( e \), making direct computation of other bases more challenging without this formula.

In our example, we used this formula to transform \( \log_{1/2} 5 \) into a form involving natural logarithms for easy calculation: \( \frac{\ln 5}{\ln (1/2)} \). This allows utilizing the richness of natural logarithms in solving the problem.

Approximation Methods

When calculating logarithmic expressions, especially with bases other than 10 or \( e \), we often rely on approximation methods to find precise numerical values. The use of calculators is commonplace to approximate values of natural logarithms such as \( \ln 5 \approx 1.6094 \) and \( \ln (1/2) \approx -0.6931 \).

Dividing the calculated natural logarithms, as in our example, gives an approximate value for the original logarithm: \( \frac{\ln 5}{\ln (1/2)} \approx -2.3219 \).

These approximations to four decimal places are crucial in fields like science and engineering, where precision is vital. By using approximation methods, we ensure that solutions are both practical and useful for real-world application, maintaining a balance between exactness and feasibility.