Problem 61 - 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. $$ \log _{b} \sqrt{5} $$

Step-by-Step Solution

Answer

The value of \( \log_{b} \sqrt{5} \) is 0.3495.

Step 1: Express the Square Root as an Exponent

Recall that a square root can be expressed as an exponent of one-half. Therefore, we can write \( \sqrt{5} \) as \( 5^{1/2} \). The expression now becomes \( \log_{b}(5^{1/2}) \).

Step 2: Use Logarithm Power Rule

The logarithm power rule states that \( \log_{b}(a^{c}) = c \cdot \log_{b}(a) \). Apply this rule to \( \log_{b}(5^{1/2}) \) to get \( \frac{1}{2} \cdot \log_{b}(5) \).

Step 3: Substitute the Given Logarithm Value

Using the given \( \log_{b}(5) = 0.6990 \), substitute this value into the expression \( \frac{1}{2} \cdot \log_{b}(5) \) to obtain \( \frac{1}{2} \cdot 0.6990 \).

Step 4: Calculate the Result

Perform the multiplication to find the value: \( \frac{1}{2} \cdot 0.6990 = 0.3495 \).

Step 5: Round the Answer

Since the problem asks for the answer to be rounded to four decimal places, the final answer is \( 0.3495 \). Ensure the result has four decimal places, which it already does.

Key Concepts

Logarithm Power Rule

When working with logarithms, you'll often encounter expressions raised to a power. Here, the logarithm power rule becomes highly useful. This rule states that if you have a logarithm of a number raised to an exponent, it can be simplified. The rule is expressed in this way: \( \log_{b}(a^{c}) = c \cdot \log_{b}(a) \). Simply put, you can "move" the exponent in front of the logarithm as a multiplier.

This significantly simplifies calculations. Instead of dealing with complex powers inside the log expression, the exponent extraction allows working with simple arithmetic.

• For example, \( \log_{b}(5^{1/2}) \) becomes \( \frac{1}{2} \cdot \log_{b}(5) \). This showcases how the power rule breaks down the complexity.

The key benefit of the logarithm power rule is reducing the steps in calculations without altering the base or the logarithmic nature of the expression. Remember, this rule applies to any real exponent, giving you flexibility in numerous situations.

Square Root as Exponent

A useful mathematical trick is expressing a square root as an exponent. This approach not only aids in algebraic manipulation but also in simplifying logarithmic expressions.

In general, the square root of a number can be represented as that number to the power of one-half: \( \sqrt{a} = a^{1/2} \). This expression brings the concept of roots into the realm of exponents, making tasks like differentiation or integration more straightforward.

• For the problem involving \( \log_{b} \sqrt{5} \), the square root expression is rewritten as \( \log_{b} (5^{1/2}) \).

This transformation aligns perfectly with the logarithm power rule, making further calculations easier. By rephrasing square roots as exponents, much of the complexity in logs and algebra can be streamlined, allowing you to solve problems with greater ease and accuracy.

Base b Logarithm

In dealing with logarithms, the base is a critical component. When you see \( \log_{b}(a) \), \( b \) is the base. The base of the logarithm determines the "scale" or "size" of the number in the logarithmic system being used.

For example, if \( \log_{b} 4 \) or \( \log_{b} 5 \) are parts of your expression, having a constant base \( b \) allows you to apply logarithmic rules consistently across similar bases.

• This is exactly why we need the base when interpreting numbers like \( 0.6021 \) or \( 0.6990 \), as provided in problems.

With a known base, these numbers become reference points you can replace into expressions to solve more complex logarithmic equations.

Furthermore, keeping track of your base not only helps ensure accuracy but also aids in converting between different log systems if necessary. In many textbook exercises, like those using formed constants as seen here, maintaining the same base is crucial to achieving the correct answer.