Problem 63 - Precalculus with Calculus Previews

Question

In Problems \(57-68\), use \(\log _{b} 4=0.6021\) and \(\log _{b} 5=\) \(0.6990\) to evaluate the given logarithm. Round your answer to four decimal places. $$ \log _{b} \sqrt[3]{4} $$

Step-by-Step Solution

Answer

The value of \(\log_b \sqrt[3]{4}\) is 0.2007.

Step 1: Rewrite the Logarithm Using the Property of Roots

The logarithm of a cube root can be rewritten using the property \(\log_b \sqrt[3]{x} = \frac{1}{3} \log_b x\). This gives us \(\log_b \sqrt[3]{4} = \frac{1}{3} \log_b 4\).

Step 2: Substitute the Given Logarithm Value

Substitute the given value of \(\log_b 4 = 0.6021\) into the expression from Step 1. This yields \(\log_b \sqrt[3]{4} = \frac{1}{3} \times 0.6021\).

Step 3: Calculate the Final Answer Using Arithmetic

Perform the arithmetic calculation \(\frac{1}{3} \times 0.6021 = 0.2007\). Thus, \(\log_b \sqrt[3]{4} = 0.2007\).

Key Concepts

Cube Root

The cube root of a number is the value that, when multiplied by itself three times, equals the original number. For example, the cube root of 8 is 2, because

• 2 × 2 × 2 = 8.

We represent the cube root of a number using the radical symbol with a small 3, such as \[ \sqrt[3]{x}. \]This notation helps simplify expressions and equations containing roots.

Understanding how cube roots transform within logarithmic functions is essential when solving logarithmic equations. Specifically, the property of logarithms help us convert the logarithm of a cube root into a more manageable expression. Using the formula:\[\log_b \sqrt[3]{x} = \frac{1}{3}\log_b x. \]This formula allows us to break down complex expressions into simpler calculations, which can then be evaluated more easily.

Logarithmic Equations

Logarithmic equations are expressions involving logarithms, which are the inverse operations of exponentiation. They help us solve problems where the unknown quantity is an exponent. This is vital when dealing with complex exponentials and roots.

In solving logarithmic equations, we often use specific properties of logarithms to simplify and manipulate expressions. For instance:

• The product property: \( \log_b (xy) = \log_b x + \log_b y \)
• The quotient property: \( \log_b \left(\frac{x}{y}\right) = \log_b x - \log_b y \)
• The power property: \( \log_b (x^n) = n \log_b x \)

These properties enable us to transform logarithmic forms, making it easier to insert known values or further simplify. They are especially useful when paired with calculations like those needed from the given log value, such as converting \( \log_b \sqrt[3]{4} \) using the power property for roots.

Arithmetic Calculations

Arithmetic calculations involve performing basic mathematical operations such as addition, subtraction, multiplication, and division. In problems involving logarithms, arithmetic plays a crucial role in determining the final answer after simplifying the expression.

After applying logarithmic properties to simplify an expression, like rewriting the cube root as \[ \frac{1}{3} \log_b 4, \]we proceed to the arithmetic calculation step: multiplying by 1/3. This step is crucial, as it finalizes the solution, yielding the ultimate value of the expression:

• \(0.2007 = \frac{1}{3} \times 0.6021\)

Precision in arithmetic calculations ensures accuracy in the solution, particularly in logarithmic equations where answer precision is often required to several decimal places.