Problem 28 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 19-34: Solve the equation. $$ \log _{4} x=-\frac{3}{2} $$

Step-by-Step Solution

Answer

The solution is \( x = \frac{1}{8} \).

Step 1: Understand the Logarithm Equation

The given equation is \( \log_{4} x = -\frac{3}{2} \). This means that we are looking for a number \( x \) such that \( 4 \) raised to the power of \( -\frac{3}{2} \) equals \( x \).

Step 2: Convert Logarithmic to Exponential Form

Rewrite the logarithmic equation \( \log_{4} x = -\frac{3}{2} \) in its equivalent exponential form: \( x = 4^{-\frac{3}{2}} \). This correlates the base, exponent, and result in a straightforward manner.

Step 3: Simplify the Negative Exponent

The expression \( 4^{-\frac{3}{2}} \) indicates the reciprocal of \( 4^{\frac{3}{2}} \). Thus, \( x = \frac{1}{4^{\frac{3}{2}}} \).

Step 4: Evaluate the Expression

Evaluate \( 4^{\frac{3}{2}} \). Break it down as \( (4^{\frac{1}{2}})^{3} = (\sqrt{4})^{3} = 2^{3} \). Therefore, \( 4^{\frac{3}{2}} = 8 \).

Step 5: Obtain the Final Solution

Substitute back the evaluated expression into \( x = \frac{1}{4^{\frac{3}{2}}} \), gives us \( x = \frac{1}{8} \). Thus, the solution is \( x = \frac{1}{8} \).

Key Concepts

Understanding Exponential Form

When dealing with logarithms, a key step is converting them into their exponential form. This process allows you to understand what the logarithm is actually expressing. A logarithm such as \( \log_{4} x = -\frac{3}{2} \) can be translated into an exponential expression: \( x = 4^{-\frac{3}{2}} \).
This conversion involves recognizing that the logarithm asks the question: "To what power must the base 4 be raised, to result in \( x \)?"
Here, the answer is \(-\frac{3}{2}\), which clarifies that \( 4 \) raised to this power equals \( x \).
Understanding this relationship is crucial:

• The base of the logarithm becomes the base of the power.
• The answer to the logarithm is the exponent.
• The result of the exponential expression is the value of \( x \).

Converting logarithmic equations into exponential form often simplifies them, making it easier to solve and to understand the mathematical relationships involved.

Decoding Negative Exponents

Negative exponents can seem tricky at first, but they are manageable once understood.
In mathematics, a negative exponent indicates a reciprocal. For instance, \( 4^{-\frac{3}{2}} \) means \( \frac{1}{4^{\frac{3}{2}}} \). It's a way of expressing division using exponents.
This makes it easier to work with large and small numbers by simplifying expressions. Here's how it is generally approached:

• Recognize that a negative exponent implies the inverse or the reciprocal.
• Re-write the expression: If you have \( a^{-n} \), convert it to \( \frac{1}{a^n} \).
• The concept helps in breaking down complex power functions into simpler fractional parts.

By breaking down the expression \( 4^{-\frac{3}{2}} \) into its reciprocal form, you can then further simplify it by evaluating the inner expression \( 4^{\frac{3}{2}} \), making calculations much more straightforward.

Evaluating Expressions with Fractional Exponents

Evaluating expressions with fractional exponents might seem complex, but they are rather straightforward when approached correctly. A fractional exponent like \( \frac{3}{2} \) can be understood in steps:

• The denominator indicates a root. So, \( \frac{1}{2} \) reflects the square root.
• The numerator indicates the power to which the number is raised after taking the root. Thus, \( 4^{\frac{3}{2}} = (\sqrt{4})^3 \).

Let's break down \( 4^{\frac{3}{2}} \):
The first step is finding the square root of 4, which is 2. Then you raise this result to the power of 3, giving \( 2^3 = 8 \).
This method simplifies the process of working with powers and roots, enabling clearer calculations and understanding.