Problem 19 - Calculus Early Transcendentals: Pearson New International Edition

Question

$$ \text { In Problems 17-24, solve for } x . \text { Hint: } \log _{a} b=c \Leftrightarrow a^{c}=b \text {. } $$ $$ \log _{4} x=\frac{3}{2} $$

Step-by-Step Solution

Answer

x = 8

Step 1: Identify the given logarithmic expression

The problem provides the logarithmic equation \( \log_{4} x = \frac{3}{2} \). We need to solve for \( x \).

Step 2: Apply the hint transformation

According to the hint provided, \( \log_{a} b = c \) implies that \( a^{c} = b \). Here, \( a = 4 \), \( c = \frac{3}{2} \), and \( b = x \).

Step 3: Rewrite the equation in exponential form

Using the transformation from Step 2, the equation becomes \( 4^{\frac{3}{2}} = x \).

Step 4: Simplify the exponential expression

Calculate \( 4^{\frac{3}{2}} \). The expression \( 4^{\frac{3}{2}} \) can be broken down as \( \left(4^{\frac{1}{2}}\right)^3 \). Since \( 4^{\frac{1}{2}} = \sqrt{4} = 2 \), we have \( 2^3 = 8 \).

Step 5: Conclude the solution

This simplification shows that \( x = 8 \), which is the solution to the equation.

Key Concepts

Exponential Form

To grasp the concept of exponential form, let's first understand what it entails. An exponential form is a way of expressing numbers using bases and exponents. In mathematics, this is often written as \( a^b \), where \( a \) refers to the base, and \( b \) is the exponent. The base represents the number that is multiplied by itself, and the exponent indicates how many times it is multiplied. When converting from a logarithmic expression, which often looks like \( \log_a b = c \), to exponential form, we rearrange it to \( a^c = b \). This is a crucial skill as it acts as a bridge from the logarithmic world to the exponential. It allows us not only to solve logarithmic equations but also to conceptualize relationships between quantities in different mathematical contexts.
Understanding this conversion is key because it transforms a problem into a more familiar format to work with, especially if you're apt at dealing with powers and roots. For instance, in our problem, \( \log_4 x = \frac{3}{2} \), by applying the conversion, gives us \( 4^{\frac{3}{2}} = x \).

Solve for X

The main goal when dealing with equations, such as logarithmic equations, is to isolate and solve for \( x \). Once you have successfully written the logarithmic equation in exponential form, the next phase is to solve for \( x \). Solving for a variable means you want to find its value.

• In our example, after converting the logarithmic equation to the exponential form \( 4^{\frac{3}{2}} = x \), the task is to perform any necessary calculations to find the value of \( x \).
• In this specific case, simplifying \( 4^{\frac{3}{2}} \) leads us directly to the value of \( x \).

The process might involve simplifying powers, like evaluating \( 4^{\frac{3}{2}} \) as \( \left(4^{\frac{1/2}}\right)^3 = 2^3 \), which equals 8. This calculation shows that \( x = 8 \), thus idependently verifying our result with actual numbers.
Consistency in following these steps helps you become more confident in dealing with similar problems.

Mathematical Transformations

Mathematical transformations are techniques used to change one mathematical expression into another. This can help in simplifying complex problems or expressing equations in a manner more suitable for solving them. One common transformation in logarithmic equations is converting them into exponential form. This particular transformation is key in revealing the underlying relationships in equations and is often applied to solve for unknowns or to make further simplifications easier.

• Transformations enable the transition between different mathematical realms — like the move from logarithmic to exponential forms, facilitating the solving process.
• In essence, these transformations are modifications applied to perform calculations more effectively or reveal deeper insights into the expression’s structure.

The transformation from \( \log_4 x = \frac{3}{2} \) to \( 4^{\frac{3}{2}} = x \) is a simple yet an effective way to solve for \( x \). The more adept you become at recognizing when and how to apply transformations, the more skilled you'll be at tackling a wide range of mathematical problems.