Problem 84 - College Algebra with Modeling and Visualization

Question

Solve each equation. Approximate answers to four decimal places when appropriate. $$2 \log _{4} x=3.4$$

Step-by-Step Solution

Answer

\(x \approx 12.1185\)

Step 1: Rewrite the Equation Using Logarithmic properties

The given equation is \(2 \log_{4} x = 3.4\). Start by isolating the logarithm by dividing both sides by 2.\[ \log_{4} x = \frac{3.4}{2} = 1.7 \]

Step 2: Convert the Logarithmic Form to Exponential Form

Recall that the logarithmic equation \(\log_{b} a = c\) can be rewritten in exponential form as \(b^{c} = a\). For our equation, \(\log_{4} x = 1.7\) becomes:\[ 4^{1.7} = x \]

Step 3: Calculate the Exponent

Calculate the value of \(4^{1.7}\) using a calculator:\[ 4^{1.7} \approx 12.1185 \]

Step 4: Round to Four Decimal Places

Round the result to four decimal places as instructed.\[ x \approx 12.1185 \]

Key Concepts

Exponential Form

In mathematics, converting between logarithmic and exponential forms is a common task. When you have a logarithmic equation like \( \log_b a = c \), it can be restated using the exponential form as \( b^{c} = a \).
This conversion is crucial because many equations are easier to handle and solve in exponential form. For example, in solving the equation \( 2 \log_{4} x = 3.4 \), dividing by 2 gives \( \log_{4} x = 1.7 \). By recognizing the logarithmic form here, we can switch to exponential form:

• Identify the base \( b = 4 \).
• The exponent \( c = 1.7 \).

So, the equation becomes \( 4^{1.7} = x \). This direct computation can immediately give us the value of \( x \), especially with the help of a calculator.
Exponential form is a powerful tool to simplify and solve equations that might otherwise be cumbersome.

Logarithmic Properties

Understanding logarithmic properties is key to solving equations involving logarithms. Logarithms convert multiplication into addition, division into subtraction, powers into multiplication, and roots into division.
One important property utilized in the given problem is scaling the logarithm:

• If you have \( a \log_b x \), you can simplify it by moving the constant \( a \) as an external factor.

To solve the equation \( 2 \log_{4} x = 3.4 \), the first step involves using this property to isolate the logarithm. Dividing the entire equation by 2 gives \( \log_{4} x = 1.7 \).
Another common and hugely useful property is the change of base formula, which can make calculations easier by changing the logarithm to a different base. While it's not used in the original stap-by-step solution, being adept with this helps for friendly adjustments during calculations.

Rounding Numbers

Rounding numbers is an important skill in mathematics to make numbers easier to work with and interpret. Rounding to four decimal places means you look at the fifth decimal place to decide whether to round up or down.
For example, if we calculate \( 4^{1.7} \), using a calculator, we get about 12.118489...
According to the rules of rounding:

• If the number at the fifth decimal place is 5 or more, we round up.
• If it's less than 5, we round down.

In this problem, the calculation gives \( x \approx 12.1185 \), where the fifth decimal place '8' prompts us to round up. This precision ensures that our mathematical results remain consistent and clear, especially when specific decimal places need to be noted for clarity in assignments and solutions.