Problem 56 - Precalculus Mathematics for Calculus

Question

Use the Change of Base Formula and a calculator to evaluate the logarithm, rounded to six decimal places. Use either natural or common logarithms. $$\log _{5} 2$$

Step-by-Step Solution

Answer

\( \log_5 2 \approx 0.430676 \).

Step 1: Recall the Change of Base Formula

The Change of Base Formula allows us to evaluate logarithms with bases other than 10 (common logarithm) by converting them into a ratio of common logarithms. The formula is given by \( \log_b a = \frac{\log_c a}{\log_c b} \), where \( c \) is a base of our choice, either 10 or \( e \) for natural logarithms.

Step 2: Choose a Base for Calculation

For this problem, you can choose either common logarithms (base 10) or natural logarithms (base \( e \)). Here, let's use common logarithms (base 10) for the calculation. This allows us to use the standard \( \log \) function on most calculators.

Step 3: Apply the Change of Base Formula

Using the formula \( \log_5 2 = \frac{\log_{10} 2}{\log_{10} 5} \), calculate the values of \( \log_{10} 2 \) and \( \log_{10} 5 \) based on your calculator.

Step 4: Use a Calculator for Logarithm Values

Compute \( \log_{10} 2 \) and \( \log_{10} 5 \) using a calculator. These values are approximately 0.301030 and 0.698970, respectively.

Step 5: Calculate the Ratio

Divide the value obtained for \( \log_{10} 2 \) by the value obtained for \( \log_{10} 5 \): \( \frac{0.301030}{0.698970} \).

Step 6: Round to Six Decimal Places

After dividing, you get approximately 0.430676.

Step 7: Final Result

The calculated value of \( \log_5 2 \) rounded to six decimal places is 0.430676.

Key Concepts

Logarithms

Logarithms are mathematical expressions that tell us the power to which a number, known as the base, must be raised to produce a given number. For instance, if we have an expression like \( b^x = a \), then \( x \) can be written as \( \log_b a \). In simple terms, logarithms are the opposite of exponentiation. Instead of asking "what is \( 2^3 \)?", which equals 8, you're asking "what power should 2 be raised to, to get 8?" and the answer is 3, or \( \log_2 8 = 3 \). This concept is incredibly useful in solving exponential equations, as it allows us to simplify calculations involving large numbers.

• Logarithms can have different bases. The most common bases are 10 and the mathematical constant \( e \).
• The higher the base, the smaller the logarithmic value, and vice versa.

This understanding is critical in algebra and calculus, especially when dealing with exponential growth or decay problems. Logarithms help us transition between exponential expressions to linear expressions, making complex calculations manageable.

Common Logarithms

Common logarithms are logarithms that have a base of 10. They are often used in everyday calculations and scientific measurements because 10 is a convenient number base for humans, thanks to our base-10 numbering system. When you see \( \log \) without a base number specified, it usually means log base 10.

• Common logarithms are usually just written as \( \log \).
• They simplify the process of working with multiplication and division, transforming these operations into addition and subtraction respectively.

A powerful feature of common logarithms is how they allow you to perform complex calculations with ease using calculators. Because most calculators have a \( \log \) button, you can directly compute logarithms in base 10, which is practical for quickly evaluating logarithmic expressions, just like in our exercise example. This is especially handy when you don't have access to advanced mathematical tools and need to perform quick calculations.

Natural Logarithms

Natural logarithms use the constant \( e \) (approximately 2.718) as their base. This type of logarithm is prevalent in advanced mathematics, especially in areas involving calculus, because \( e \) is the base rate of growth shared by all continually compounding processes.

• The natural logarithm is often denoted as \( \ln \).
• Natural logarithms are widely used in scientific fields, particularly when modeling continuous growth or decay, such as in population dynamics or radioactive decay.

The unique properties of \( e \) make natural logarithms exceptionally useful for calculations involving complex numbers and exponential growth models. They're essential in calculus, particularly when differentiating or integrating equations involving exponential functions. This means understanding natural logarithms can greatly enhance your mathematical toolkit, especially in higher-level mathematics and applied sciences.