Problem 34 - College Algebra with Corequisite Support

Question

For the following exercises, use the change-of-base formula to evaluate each expression as a quotient of natural logs. Use a calculator to approximate each to five decimal places. \(\log _{8}(65)\)

Step-by-Step Solution

Answer

\( \log_8(65) \approx 2.00742 \)

Step 1: Recall the Change-of-Base Formula

The change-of-base formula for logarithms allows us to rewrite a logarithm in terms of logs of other bases. The formula is given by:\[\log_b(a) = \frac{\log_c(a)}{\log_c(b)}\]where \(c\) is any positive number different from 1, often chosen as 10 or \(e\), the base of natural logarithms.

Step 2: Apply the Change-of-Base Formula

To evaluate \( \log_8(65) \) using the change-of-base formula, we can express it in terms of natural logarithms:\[\log_8(65) = \frac{\ln(65)}{\ln(8)}\]This allows us to use a calculator to find these logarithm values.

Step 3: Evaluate the Natural Logs Using a Calculator

Use a calculator to find the natural logarithms:- Calculate \( \ln(65) \) which is approximately 4.17439.- Calculate \( \ln(8) \) which is approximately 2.07944.

Step 4: Divide the Logarithm Values

Using the values from Step 3, divide the natural logs to find the desired log:\[\frac{\ln(65)}{\ln(8)} = \frac{4.17439}{2.07944} \approx 2.00742\]

Step 5: Approximate the Final Answer

The approximate value of \( \log_8(65) \) to five decimal places is 2.00742.

Key Concepts

Logarithms

Logarithms are a mathematical tool used to solve equations involving exponents. At its core, a logarithm tells us how many times we multiply a base number to reach another number. For example, in the expression \( \log_8(65) \), 8 is the base, and the logarithm asks: how many 8's summed together through multiplication equal 65?
Logarithms simplify complex multiplication into straightforward addition, offering an easier approach to handling exponential problems. Key properties include:

• Product Rule: \( \log_b(M \times N) = \log_b(M) + \log_b(N) \)
• Quotient Rule: \( \log_b(M/N) = \log_b(M) - \log_b(N) \)
• Power Rule: \( \log_b(M^k) = k \times \log_b(M) \)

These rules help simplify complex logarithmic expressions into solvable forms, making calculations more manageable.

Natural Logarithms

The natural logarithm, denoted as \( \ln \), is a specific type of logarithm where "e" (approximately 2.71828) is the base. Natural logs are often used in applications involving continuous growth or compounding, such as in the fields of finance and biology.
What makes natural logarithms unique is their connection to the mathematical constant, "e," which is the base rate of growth shared by all continually growing processes. It's non-arbitrary; instead, it stems from the properties of natural growth systems.

• Natural logarithms convert multiplicative relationships into additive ones.
• They simplify equations where "e" plays a role, making them crucial in calculus and natural science applications.

In complex equations, using natural logs allows for straightforward transformations and simplifications, making problem-solving more manageable.

Calculator Usage

Calculators simplify the process of working with logarithmic expressions, especially when decimals and natural logarithms are involved. When using a calculator, ensure it's set to the correct mode and understand how to access its logarithmic functions properly.
Common steps include:

• Select the "ln" button for natural logarithms when determining values like \( \ln(65) \).
• Press the "÷" or division key to calculate expressions like \( \frac{\ln(65)}{\ln(8)} \).
• Adjust settings to ensure high precision and select a mode where decimal approximation is displayed to enough significant figures.

By gaining proficiency with calculator functions, you can confidently transition through exercises involving the Change of Base Formula and other complex calculations.

Step-by-Step Algebra Solutions

Solving log expressions step-by-step involves understanding the principles of logarithms and systematically applying them until a solution is reached. Globally recognized as one of the most methodical ways to tackle algebra problems, this approach is fundamental to mastering math concepts.
Begin by identifying the main expression or equation you need to simplify. Then, apply specific rules or formulas, like the Change of Base Formula. Write each step methodically to prevent errors, and always verify your calculations. For most math students, breaking down problems as shown:

• Use formulas, such as the change-of-base formula: \( \log_b(a) = \frac{\log_c(a)}{\log_c(b)} \)
• Convert any complex expression into a simpler, more familiar logarithmic form.
• Check all steps using a calculator to ensure your answers are precise.

Practicing these steps regularly enhances problem-solving skills and builds confidence in dealing with diverse algebraic challenges.