Problem 66 - Algebra 2 and Trigonometry

Question

In \(57-68,\) solve each equation for the variable. $$ \log _{8} x=\frac{1}{2} $$

Step-by-Step Solution

Answer

The solution is \( x = 2\sqrt{2} \).

Step 1: Identify the Equation

We have the equation given as \( \log_{8} x = \frac{1}{2} \). This equation is in logarithmic form involving base 8.

Step 2: Convert the Logarithmic Equation to Exponential Form

To solve the logarithmic equation, convert it to its corresponding exponential form. The general form \( \log_{b} a = c \) can be converted to exponential form \( b^{c} = a \). Thus, \( 8^{\frac{1}{2}} = x \).

Step 3: Simplify the Exponential Expression

Calculate \( 8^{\frac{1}{2}} \). The expression \( 8^{\frac{1}{2}} \) means the square root of 8. Simplifying this, we get \( \sqrt{8} = \sqrt{4 imes 2} = \sqrt{4} \times \sqrt{2} = 2\sqrt{2} \).

Step 4: Write the Final Solution

Thus the solution is \( x = 2\sqrt{2} \). We solved for \( x \) in terms of the simplest radical form.

Key Concepts

Exponential Form

Converting logarithmic equations to exponential form is like changing a complex problem into a simpler one. In a logarithmic equation like \( \log_{b} a = c \), you are essentially saying that the base \( b \) raised to what power equals \( a \).

• This conversion is useful because exponential equations are usually easier to solve.
• The conversion follows the rule \( b^{c} = a \), which is a fundamental concept to grasp when working with these kinds of problems.

For the original exercise \( \log_{8} x = \frac{1}{2} \), we turn it into its exponential form to make it easier to solve: \( 8^{\frac{1}{2}} = x \). This change makes calculations more straightforward, as you can now directly compute the value of \( x \).

Simplifying Radicals

Once you have an exponential expression like \( 8^{\frac{1}{2}} \), it is often helpful to simplify it further. Simplifying radicals helps in shortening the expression and making the answer easier to interpret.

• The expression \( 8^{\frac{1}{2}} \) represents the square root of 8.
• To simplify \( \sqrt{8} \), break it down into factors: \( \sqrt{4 \times 2} \). This is crucial as it identifies integers whose roots are easy to simplify.
• Calculate \( \sqrt{4} \) as 2 and leave \( \sqrt{2} \) as it is, leading to the simpler form: \( 2\sqrt{2} \).

Understanding how to simplify radicals is an essential mathematical skill that makes the solution appear in its simplest form, which is often required in mathematical problems and proofs.

Logarithmic Form

Logarithmic form is a compact way of expressing the relationship between three critical components: the base, the exponent, and the result. In the expression \( \log_{b} a = c \), the base \( b \) raised to the exponent \( c \) equals \( a \):

• This formulation is particularly useful to solve problems involving large numbers and exponents.
• In our example, \( \log_{8} x = \frac{1}{2} \), the base is 8, the result is \( x \), and the exponent \( \frac{1}{2} \) is its logarithm.

Logarithms simplify multiplication and exponentiation problems by turning them into addition and multiplication of simpler numbers. They are particularly powerful in cases involving complex calculations or scientific problems.

Solving Equations

Solving logarithmic equations involves a clear step-by-step approach, which combines the understanding of both exponential and logarithmic forms.

• Start by converting the problem into a form that is easy to manipulate, like changing a logarithmic form into an exponential one.
• Calculating factors is essential, as seen when simplifying radicals such as \( 8^{\frac{1}{2}} \), which simplifies to \( 2\sqrt{2} \).
• Verify your results by checking the computation process to ensure that each step was accurate.

The solution \( x = 2\sqrt{2} \) shows how logarithms offer a powerful tool for converting complex equations into manageable steps, allowing for effective problem-solving in mathematics.