Problem 44 - Intermediate Algebra

Question

For Problems \(41-50\), solve each equation. $$ \log _{16} x=\frac{3}{2} $$

Step-by-Step Solution

Answer

The solution is \( x = 64 \).

Step 1: Understand the equation

The equation given is \( \log_{16} x = \frac{3}{2} \). This is a logarithmic equation where the base is 16, the unknown is \( x \), and the logarithm equals \( \frac{3}{2} \).

Step 2: Convert the logarithmic equation to an exponential form

We use the property of logarithms that states if \( \log_b a = c \), then \( a = b^c \). Applying this to our equation, we get \( x = 16^{\frac{3}{2}} \).

Step 3: Simplify the exponential expression

The expression \( 16^{\frac{3}{2}} \) means the square root of 16 raised to the power of 3. First, find the square root of 16 which is 4. Then raise 4 to the power of 3 to get \( 4^3 = 64 \). Thus, \( 16^{\frac{3}{2}} = 64 \).

Step 4: Write the solution

The value of \( x \) that satisfies the original equation is \( x = 64 \). This is the solution to the problem.

Key Concepts

Exponential Form

When dealing with logarithmic equations, an essential skill is converting them into exponential form. This conversion helps in simplifying and solving the equation more comfortably. In the given logarithmic equation \( \log_{16} x = \frac{3}{2} \), the base is 16, and the result of the logarithm is \( \frac{3}{2} \). Converting this to an exponential form involves using the rule that if \( \log_b a = c \), then \( a = b^c \). Applying this rule, we rewrite the equation as \( x = 16^{\frac{3}{2}} \).

• Logarithmic to Exponential: The expression turns into an exponential equation that is generally much easier to understand and work with.
• Shorter Steps to Solve: Exponential equations can usually be simplified quickly compared to logarithmic equations.

After rewriting the logarithm in exponential terms, you can start working on simplifying the resulting expression.

Simplifying Expressions

Once you have the expression in exponential form, the next step is simplifying it. The given expression, \( 16^{\frac{3}{2}} \), represents calculating a radical and a power. Specifically, it means taking the square root of 16 and then raising the result to the power of 3. Let's break it down:

• Square Root: The square root of 16 is 4, as \( 4 \times 4 = 16 \).
• Raising to Power: After finding the square root, you then raise 4 to the power of 3. Calculating \( 4^3 \) gives you 64, as \( 4 \times 4 \times 4 = 64 \).

This two-step process of simplifying the expression yields a clean numerical value, making it easier to understand and check your solution. Simplification helps ensure you have executed the conversion from logarithmic to exponential form correctly.

Solving Equations

Finally, after simplifying the exponential expression to a single numerical value, the solution to the original equation is more apparent. In this instance, we've simplified \( 16^{\frac{3}{2}} \) to equal 64. Thus, the original equation \( \log_{16} x = \frac{3}{2} \) resolves as \( x = 64 \).

• Verification: Always double-check your work. You can do this by substituting the solution back into the original equation to ensure it holds true.
• Understanding Context: Solving these equations helps in various real-world applications, such as computing growth rates or solving problems in computer science where logarithms frequently appear.

Successfully solving the equation signifies not only getting the correct answer but also understanding the pathway to that solution, which reinforces mastery of logarithmic and exponential relationships.