Problem 44 - Algebra for College Students

Question

Solve each equation. \(\log _{16} x=\frac{3}{2}\)

Step-by-Step Solution

Answer

The solution to the equation is \( x = 64 \).

Step 1: Understand the Given Equation

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

Step 2: Convert the Logarithmic Equation to Exponential Form

To solve the logarithmic equation, we first convert it to its equivalent exponential form. In general, if \( \log_b a = c \), then \( a = b^c \). Applying this to our equation, we have \( x = 16^{\frac{3}{2}} \).

Step 3: Simplify the Exponential Expression

Now, we need to simplify \( 16^{\frac{3}{2}} \). Start by rewriting 16 as a power of 4: \( 16 = 4^2 \). Thus, \( 16^{\frac{3}{2}} = (4^2)^{\frac{3}{2}} \).

Step 4: Apply the Power Rule of Exponents

Using the power rule \( (a^m)^n = a^{m \cdot n} \), we can simplify: \((4^2)^{\frac{3}{2}} = 4^{2 \cdot \frac{3}{2}} = 4^3 \).

Step 5: Calculate the Final Answer

Finally, calculate \( 4^3 \) which equals \( 4 \times 4 \times 4 = 64 \). Thus, \( x = 64 \).

Key Concepts

Exponential Form

A key concept in solving logarithmic equations is converting them into exponential form. This step transforms a logarithmic statement into an exponential one, which is often simpler to work with. In our exercise, we begin with a logarithmic equation:

• \( \log_{16} x = \frac{3}{2} \)

This states that the logarithm of \(x\) with base 16 equals \( \frac{3}{2} \).
By translating this into exponential form, we apply the general rule \( \log_{b} a = c \implies a = b^c \).
By doing so, we find:

• \( x = 16^{\frac{3}{2}} \)

Using exponential form allows us to directly solve for \(x\) by exponentiation, making the equation more manageable.

Power Rule of Exponents

When working with exponents, one powerful tool is the power rule of exponents. This rule simplifies exponential expressions that involve a power raised to another power. The general form of this rule is:

• \( (a^m)^n = a^{m\cdot n} \)

In the given problem, we start with the expression:

• \( 16^{\frac{3}{2}} \)

Recognizing 16 as a power of 4, specifically \( 4^2 \), allows us to express this as:

• \((4^2)^{\frac{3}{2}} \)

Applying the power rule gives us:

• \((4^2)^{\frac{3}{2}} = 4^{2 \cdot \frac{3}{2}} = 4^3 \)

This simplification is crucial for finding the exact value of the expression, streamlining the computation process.

Simplifying Expressions

Simplifying expressions involves making them easier to understand and work with. For exponential equations, this often means rewriting components to enable easier computation or further simplification.
In the example, we begin with:

• \( x = 16^{\frac{3}{2}} \)

Since 16 is \( 4^2 \), we express it in a form that takes advantage of the power rule. Once transformed to \( (4^2)^{\frac{3}{2}} \), applying the power rule is straightforward, simplifying it to \( 4^3 \).
Once at \( 4^3 \), calculating:

• \( 4 \times 4 \times 4 = 64 \)

Completes the simplification process. Simplifying thoroughly makes complex-looking expressions easy to evaluate and confirms that our solutions are correct.