Problem 17 - Algebra A Combined Function

Question

Solve each equation. $$ \log _{4} 2+\log _{4} x=0 $$

Step-by-Step Solution

Answer

The solution is \( x = \frac{1}{2} \).

Step 1: Use the Logarithm Addition Property

We start with the equation: \( \log_{4} 2 + \log_{4} x = 0 \). By applying the logarithm addition property, \( \log_{a} b + \log_{a} c = \log_{a}(b \cdot c) \), we combine the logs: \( \log_{4} (2x) = 0 \).

Step 2: Convert Logarithmic Equation to Exponential Form

Next, we need to convert the logarithmic equation \( \log_{4} (2x) = 0 \) into its exponential form. By definition, if \( \log_{a} b = c \), then \( a^c = b \). Therefore, \( 4^0 = 2x \).

Step 3: Solve the Exponential Equation

We have the equation \( 4^0 = 2x \). Since \( 4^0 = 1 \), we have \( 1 = 2x \). Dividing both sides by 2 gives us \( x = \frac{1}{2} \).

Key Concepts

Logarithm Addition Property

When dealing with logarithms, the addition property is a handy tool that helps simplify equations. If you encounter an equation like \( \log_{a} b + \log_{a} c \), you can combine the two logs by using the property that states:

• \( \log_{a} b + \log_{a} c = \log_{a}(b \cdot c) \)

This means you can multiply the arguments (\(b\) and \(c\), in this case) into a single logarithm.

For example, in the exercise \( \log_{4} 2 + \log_{4} x = 0 \), you can merge the logs into:

\( \log_{4} (2x) = 0 \).

Knowing how to use the logarithm addition property allows you to simplify complex logarithmic expressions, making it easier to solve them.

Exponential Form

Converting a logarithmic equation to its exponential form is a crucial step in solving it. The logarithmic form \( \log_{a} b = c \) can be rewritten as an exponential equation:

• \( a^c = b \)

This transformation is powerful because it turns the equation into a form that's often easier to manipulate and solve.

In our example, after using the logarithm addition property, we end up with the equation \( \log_{4} (2x) = 0 \). By converting into exponential form, we express this as:

\( 4^0 = 2x \).

Understanding exponential form is key, as it reveals the corresponding relationship between the components of the logarithm, thereby simplifying the path to find the solution.

Solving Equations

Once an equation is simplified and transformed, solving it becomes a straightforward process. For the exponential form \( 4^0 = 2x \), it helps to remember that any non-zero number raised to the power of zero is 1:

• \( 4^0 = 1 \)

This step simplifies our equation to \( 1 = 2x \), making it easy to solve for \(x\).

To isolate \(x\), divide both sides of the equation by 2, resulting in:

• \( x = \frac{1}{2} \)

Solving equations is about systematically reducing them to their simplest form, allowing you to uncover the unknown variable. Through steps like applying the addition property and converting to exponential form, you make the equations easier to handle.