Problem 5 - Precalculus: Functions and Graphs

Question

Estimate using the change of base formula. $$\log _{5} 6$$

Step-by-Step Solution

Answer

\(\log_5 6 \approx 1.113\) using base 10.

Step 1: State Change of Base Formula

The change of base formula allows us to convert logarithms with any given base to logarithms with another base (usually 10 or e). The formula is:\[ \log_a b = \frac{\log_c b}{\log_c a} \]where \(a\) is the original base, \(b\) is the argument of the logarithm, and \(c\) is the new base.

Step 2: Choose New Base

Common choices for the new base \(c\) are \(10\) (common logarithm) or \(e\) (natural logarithm). In this exercise, we will use \(c = 10\).

Step 3: Apply the Change of Base Formula

We apply the change of base formula to convert \(\log_5 6\):\[\log_5 6 = \frac{\log_{10} 6}{\log_{10} 5}\]This formula will allow us to use a calculator to find the approximate value of \(\log_5 6\).

Step 4: Use a Calculator to Find Logs

Using a calculator, compute:- \(\log_{10} 6 \approx 0.7782\)- \(\log_{10} 5 \approx 0.6990\)

Step 5: Divide the Logarithms

Now, divide the logarithms obtained in Step 4:\[\frac{\log_{10} 6}{\log_{10} 5} = \frac{0.7782}{0.6990} \approx 1.113\]Thus, \(\log_5 6 \approx 1.113\).

Key Concepts

Logarithms

Logarithms are the inverse operation of exponentiation, much like how subtraction is the inverse of addition. They can help you figure out what power you need to raise a base number to, in order to get another number. For example, if you know that

• \( 2^3 = 8 \)

you can say that the logarithm of 8 with base 2 is 3, expressed as \( \log_2 8 = 3 \). Logarithms have certain properties that make them really useful in math. Some key properties include:

• Product Rule: \( \log_b (mn) = \log_b m + \log_b n \)
• Quotient Rule: \( \log_b \left( \frac{m}{n} \right) = \log_b m - \log_b n \)
• Power Rule: \( \log_b (m^n) = n \cdot \log_b m \)

These rules can help you simplify and solve logarithms in various situations.When estimating logarithms, especially with uncommon bases, it’s often helpful to use the change of base formula. This assists by allowing you to use a calculator to find your logarithm in more familiar bases like 10 or \( e \).

Common Logarithm

A common logarithm has a base of 10 and is denoted as \( \log_{10} \). However, it's often just written as \( \log \) without the base shown. This is because using 10 as a base is so common, especially in scientific and technical fields.When you see \( \log \), you can assume it’s a base 10 logarithm, known for its utility with decimal and scientific notation:

• Common logarithms are particularly convenient for solving equations involving powers of 10.
• They are integrated into many calculators, making it easy to carry out calculations quickly.
• One practical use is in determining the pH of solutions, with the formula \( \text{pH} = -\log [H^+] \), where \([H^+]\) is the concentration of hydrogen ions.

Using the change of base formula, as illustrated in the solution, allows you to find logarithms of different bases using common logarithm values, simplifying the calculations by drawing on the widespread availability of \( \log \).

Natural Logarithm

The natural logarithm is another special type of logarithm, and it has the base \( e \), an irrational constant approximately equal to 2.718. The natural logarithm is denoted as \( \ln \). Natural logarithms are used extensively in higher-level mathematics and calculus. Some reasons for their importance include:

• Growth and Decay Models: Natural logarithms are useful for modeling exponential growth and decay processes, such as population growth or radioactive decay.
• Calculus: Natural logarithms are linked with the derivative and integral of exponential functions. The function \( e^x \) has a derivative of \( e^x \), and its integral is \( \ln(x) \), showcasing the intertwined nature of \( e \) and logarithms.
• Finance: Compound interest calculations often involve \( e \), since \( e \) represents continuously compounded interest.

In the context of the change of base formula, natural logarithms can also be used as the base \( c \), providing another method for solving logarithms with less common bases. Understanding natural logarithms paves the way for deeper applications in scientific fields.