Problem 81 - College Algebra with Modeling and Visualization

Question

Solve each equation. Approximate answers to four decimal places when appropriate. $$\log _{2} x=1.2$$

Step-by-Step Solution

Answer

x \approx 2.2974

Step 1: Identifying the Equation Type

The given equation is \( \log_{2} x = 1.2 \). This is a logarithmic equation where the base of the logarithm is 2.

Step 2: Converting Logarithmic to Exponential Form

To solve a logarithmic equation, convert it into its equivalent exponential form. The general form of a logarithmic equation \( \log_b a = c \) can be converted to its exponential form \( b^c = a \). For the given problem, \( \log_2 x = 1.2 \) can be rewritten as \( 2^{1.2} = x \).

Step 3: Calculating the Exponential Value

Now, calculate \( 2^{1.2} \) to find the value of \( x \). Using a calculator, \( 2^{1.2} \) is approximately equal to 2.2974.

Key Concepts

Exponential Form

Understanding exponential form is crucial while working with logarithmic equations. When you have a logarithmic equation such as \( \log_b a = c \), transforming it into exponential form involves rearranging the equation. You shift the base \( b \) with the power \( c \) to equal the result \( a \). This conversion is invaluable because it simplifies the equation into a format that's easy to compute. For example, taking our equation \( \log_2 x = 1.2 \), we convert it to the form \( 2^{1.2} = x \). This reveals that \( x \) is a power of 2, specifically \( 2 \) raised to the power of 1.2. With this adjustment, what initially looked complex becomes a more straightforward calculation task. Breaking it down this way helps demystify logarithmic problems, turning them into manageable steps.

Base Conversion

Base conversion, especially when dealing with logarithms, is about changing the base to simplify solving equations. When dealing with a logarithmic equation such as \( \log_2 x = 1.2 \), base 2 is used naturally in calculations, often directly leading into exponential form without additional conversion need. However, understanding base changes helps in cases where simplification or comparisons of growth rates between different bases are required. Sometimes, you'll encounter logarithms with bases like 10, commonly used in logarithm tables and calculators due to their simplicity in the decimal system.

• Ensure the base is positive and not equal to one, as these conditions make the logarithm undefined.
• Convert between bases for comparative analysis, though in most direct computations calculators will manage it seamlessly.

Grasping changes in bases strengthens comprehension, but for digital computations—like our example—you usually rely on the given base.

Calculator Usage

Using a calculator is essential when solving exponential forms of equations, especially when approximations are needed. In our equation \( 2^{1.2} = x \), conversion from logarithmic to exponential form prepares our value for direct calculator computation. Here's how to make the most out of your calculator:

• Power Function: Most scientific calculators have a power function, often marked as \( \hat{} \) or \( y^x \). Use this to compute \( 2^{1.2} \).
• Precision: Input carefully to ensure accuracy, and follow the calculator instructions exactly to arrive at approximate values.
• Rounding: For homework or test answers, round your results to four decimal places, as indicated—here, resulting approximately in 2.2974.

Effective calculator use bridges the gap between complex mathematics and practical results, making problem-solving efficient and precise.