Problem 50 - 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. $$ \log _{5} 2 $$

Step-by-Step Solution

Answer

\( \log_5 2 \approx 0.430677 \)

Step 1: Identify the Change of Base Formula

The Change of Base Formula allows you to convert a logarithm of any base to a logarithm of a different base using the expression \( \log_{b} a = \frac{\log_{c} a}{\log_{c} b} \), where \( c \) is the new base, typically 10 (common logarithm) or \( e \) (natural logarithm).

Step 2: Apply the Change of Base Formula

Choose either common logarithms or natural logarithms to apply the formula. For this example, let's use the common logarithm (base 10). The conversion is \( \log_{5} 2 = \frac{\log_{10} 2}{\log_{10} 5} \).

Step 3: Calculate the Logarithms

Use a calculator to find \( \log_{10} 2 \) and \( \log_{10} 5 \). Enter these calculations separately into your calculator. You should find that \( \log_{10} 2 \approx 0.301030 \) and \( \log_{10} 5 \approx 0.698970 \).

Step 4: Compute the Result

Divide the common logarithm of 2 by the common logarithm of 5 using the values obtained: \( \frac{0.301030}{0.698970} \approx 0.430677 \).

Step 5: Round to Six Decimal Places

Ensure that you have rounded the final result correctly to six decimal places. In this case, the evaluated logarithm is \( 0.430677 \).

Key Concepts

Logarithms

Logarithms are mathematical operations that help us solve problems involving exponentiation. They find the power a base number should be raised to obtain a given value. When we write \( \log_b a = x \), we mean \( b^x = a \). In this context, \( b \) is the base, \( a \) is the value we're interested in, and \( x \) is the exponent.
Logarithms have various properties and rules that make complex calculations simpler. These rules include the product, quotient, and power rules. They allow conversion of multiplication into addition, division into subtraction, and powers into multiplication. Logarithms are also crucial in various fields such as science, engineering, and finance for solving exponential growth or decay problems. In these scenarios, logarithms help in changing non-linear relationships into linear forms, making them easier to analyze.

Common Logarithm

A common logarithm is a type of logarithm that uses base 10. It is typically represented as \( \log \) without specifying the base, although sometimes you may see it denoted as \( \log_{10} \). Common logarithms are especially useful because our number system is based on powers of 10.
Using common logarithms simplifies many computations and aids in mental math. For example, to find the amplitude of an earthquake or the pH value of a solution, scientists use logarithmic scales based on 10. When evaluating logarithms such as \( \log_{10} 2 \) or \( \log_{10} 5 \), calculators provide quick access to these values, making numerical calculations efficient.
The Change of Base Formula highlights a practical scenario where common logarithms come in handy, allowing users to convert logarithms to a more familiar base (10 in this case), easily interpretable and computable with standard calculators.

Natural Logarithm

Natural logarithms use base \( e \), where \( e \approx 2.71828 \), a mathematical constant that is prevalent in natural growth processes. These logarithms are denoted as \( \ln \) rather than \( \log \), and are indispensable in calculus and various scientific fields because of their unique derivative and integral properties.
The natural logarithm is fundamental in equations involving continuous growth or decay, such as population growth models or radioactive decay. Equations that involve the natural exponential function \( e^x \) frequently require manipulation using natural logarithms to solve for \( x \).
When applying the Change of Base Formula, natural logarithms can be chosen as the new base \( c \), which yields the expression \( \log_5 2 = \frac{\ln 2}{\ln 5} \). When using natural logarithms, like common logarithms, calculations can be made efficiently with a scientific calculator.