Problem 58 - Algebra A Combined Function

Question

If \(\log _{b} 3=0.5\) and \(\log _{b} 5=0.7,\) evaluate each expression. $$ \log _{b} \sqrt[4]{3} $$

Step-by-Step Solution

Answer

The value of \( \log_{b} \sqrt[4]{3} \) is 0.125.

Step 1: Understand the Expression

The given expression to evaluate is \( \log_{b} \sqrt[4]{3} \). It involves a fourth root.

Step 2: Convert Root to Exponent Form

We know that the fourth root of a number can be expressed as a fractional exponent. Therefore, \( \sqrt[4]{3} \) is the same as \( 3^{\frac{1}{4}} \).

Step 3: Apply the Power Rule of Logarithms

The power rule for logarithms states that \( \log_{b} (a^{n}) = n \cdot \log_{b} a \). Applying this rule, we get \( \log_{b} (3^{\frac{1}{4}}) = \frac{1}{4} \cdot \log_{b} 3 \).

Step 4: Substitute the Known Logarithm Value

We know from the problem statement that \( \log_{b} 3 = 0.5 \). Substituting this value in, we have \( \frac{1}{4} \log_{b} 3 = \frac{1}{4} \times 0.5 \).

Step 5: Perform the Calculation

Calculate \( \frac{1}{4} \times 0.5 = 0.125 \). This is the evaluated value of the expression.

Key Concepts

Power Rule of Logarithms

The power rule of logarithms is a handy tool when you deal with exponential expressions inside logarithms. The rule states that

• \( \log_{b}(a^{n}) = n \cdot \log_{b} a \)

In simpler terms, this means you can take the exponent (power) of a term and move it in front of the log as a multiplier.
This rule is particularly useful for evaluating expressions like \( \log_{b}(3^{\frac{1}{4}}) \).
By bringing the fractional exponent out in front, you simplify the calculation significantly.
Here's a quick breakdown of why this works:

• Think of exponents in multiplication terms: \( a^{n} = a \times a \times a \ldots \) (n times).
• Logarithms allow us to transform multiplication into addition, so when you have an exponent, it's essentially a repeated addition.

With the power rule, you reduce the complexity of such repeated operations by dealing with the exponent first, thereby simplifying the logarithmic expression.

Fractional Exponents

Fractional exponents are a way to express roots as powers. This is useful because it unifies the notion of taking roots and powers into a single framework, making calculations easier.

• The expression \( \sqrt[4]{3} \) can be written as \( 3^{\frac{1}{4}} \).
• This is because the denominator of the exponent tells you the root to be taken (in this case, 4 indicates the fourth root).

Using fractional exponents allows us to apply logarithmic rules cleanly.
When solving

• \( \log_{b}(\sqrt[4]{3}) \),
• it's crucial to first convert the root into a fractional exponent, \( 3^{\frac{1}{4}} \),
• Then apply the power rule to make further simplifications.

This method simplifies complex expressions and fits them into the rules of exponentiation and logarithms smoothly, allowing for straightforward calculations.

Evaluating Logarithms

Evaluating logarithms can often seem daunting, yet it's straightforward once you grasp the necessary steps and rules.

• In our problem, we've been given \( \log_{b} 3 = 0.5 \).
• The task is to evaluate \( \log_{b}(\sqrt[4]{3}) \).

Begin by rewriting the root using fractional exponents as \( \log_{b}(3^{\frac{1}{4}}) \).
Next, apply the power rule, which simplifies the expression to \( \frac{1}{4} \cdot \log_{b} 3 \).
Finally, substitute the known value, yielding:

• \( \frac{1}{4} \cdot 0.5 = 0.125 \)

By breaking down each step, evaluating logarithms becomes a more manageable task.
Note that while it might seem repetitive, these systematic transformations make sure you're on the right path and aid in developing intuitive problem-solving skills for logarithmic expressions.