Problem 47 - Intermediate Algebra

Question

For Problems \(41-50\), solve each equation. $$ \log _{4} x=-\frac{3}{2} $$

Step-by-Step Solution

Answer

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

Step 1: Identify the Logarithmic Equation

The equation given is a logarithmic equation: \( \log _{4} x = -\frac{3}{2} \). This means that we need to find the value of \( x \) such that when 4 is raised to the power \(-\frac{3}{2}\), it equals \( x \).

Step 2: Convert to Exponential Form

Rewrite the logarithmic equation in its exponential form. This means solving for \( x \) so that \( x = 4^{-\frac{3}{2}} \). To do this, recall that \( \log _{b} a = c \) is equivalent to \( a = b^c \).

Step 3: Simplify the Exponent

Evaluate \( 4^{-\frac{3}{2}} \). We can do this by first understanding \( -\frac{3}{2} \) as \( -(\frac{3}{2}) \). This means we find the square root of 4 first and then raise it to the power 3. Hence, \( 4^{-\frac{3}{2}} = \left( \sqrt{4} \right)^{-3} = (2)^{-3} \).

Step 4: Calculate the Final Solution

Now compute \( (2)^{-3} \), which is the same as \( \frac{1}{2^3} \). This evaluates to \( \frac{1}{8} \). Therefore, \( x = \frac{1}{8} \).

Key Concepts

Exponential Form

When we deal with logarithmic equations like \( \log_{b} a = c \), we need to convert these equations into their exponential form to easily find the unknown values. The conversion is straightforward: \( a = b^c \). This change usually simplifies the equation. It involves using the base of the logarithm, which in our exercise is 4, raising it to the power on the other side of the equation, which is \(-\frac{3}{2}\). Thus, \( \log_{4} x = -\frac{3}{2} \) becomes \( x = 4^{-\frac{3}{2}} \). Understanding how to rearrange a logarithmic equation into an exponential form is like lifting a veil, revealing the solution path clearly. Always remember this step when stuck with logarithmic expressions. A quick tip: consider a logarithm as asking the question, "To what power must the base be raised to produce a particular number?" In converting, you answer that question by using the given power.

Logarithm Rules

To maneuver successfully through logarithmic equations, we need to become familiar with basic logarithm rules. These rules make solving such equations a methodical task rather than a confusing puzzle.Here are some helpful rules:

• \( \log_{b} 1 = 0 \) : because any number raised to the power of 0 is 1.
• \( \log_{b} b = 1 \) : because the base raised to the power of 1 is itself.
• \( \log_{b} (mn) = \log_{b} m + \log_{b} n \) : the logarithm of a product is the sum of the logarithms.
• \( \log_{b} \left(\frac{m}{n}\right) = \log_{b} m - \log_{b} n \) : the logarithm of a quotient is the difference of the logarithms.
• \( \log_{b} (m^n) = n \cdot \log_{b} m \) : the logarithm of a power is the exponent times the logarithm of the base.

These rules pave the way for simplifying and solving equations. Each rule can offer a different perspective or approach depending on the form of the logarithmic equation at hand.

Solving Equations

Solving logarithmic equations can be a step-by-step process similar to solving any other type of equation, but with specific operations that apply to logarithms and exponents. The ultimate goal is to isolate the variable. Here's a simple breakdown of steps you can follow:

• **Convert to Exponential Form:** As discussed, this is the first step if the equation is in the logarithmic form.
• **Simplify the Exponential Expression:** Use your knowledge of exponents to simplify. For instance, simplifying \(4^{-\frac{3}{2}}\) involves finding the square root of 4, then raising it to the 3rd power and applying the negative exponent rule to get \( \frac{1}{8} \).
• **Solve for the Variable:** With the equation simplified, you can now easily solve for \( x \).

When you methodically tackle each part of the equation, especially in understanding the transition from logarithmic to exponential form, daunting math begins to feel manageable. The key is practice and a good grasp of the logarithmic and exponential operations.