Problem 28 - Algebra and Trigonometry

Question

\(25-32\) Use the definition of the logarithmic function to find \(x\). $$ \begin{array}{ll}{\text { (a) } \log _{4} 2=x} & {\text { (b) } \log _{4} x=2}\end{array} $$

Step-by-Step Solution

Answer

(a) \( x = \frac{1}{2} \), (b) \( x = 16 \)

Step 1: Define the logarithmic function for (a)

The equation given is \( \log_{4} 2 = x \). By definition, a logarithmic equation \( \log_{b} a = c \) is equivalent to the exponential equation \( b^c = a \). Therefore, we can rewrite \( \log_{4} 2 = x \) as \( 4^x = 2 \).

Step 2: Solve the exponential equation for (a)

Now that we have the equation \( 4^x = 2 \), realize that 2 can be expressed as a power of 4. This gives us \( 2 = 4^{1/2} \), since the square root of 4 is 2. Thus, we have \( 4^x = 4^{1/2} \).

Step 3: Equate the exponents for (a)

Since the bases are the same \( (4) \), we can equate the exponents from \( 4^x = 4^{1/2} \). This gives us \( x = 1/2 \).

Step 4: Define the logarithmic function for (b)

The equation given is \( \log_{4} x = 2 \). Again, using the logarithmic definition, we rewrite it as the exponential equation: \( 4^2 = x \).

Step 5: Solve the exponential equation for (b)

Calculate \( 4^2 \), which equals \( 16 \). Thus, \( x = 16 \).

Key Concepts

Exponential Equations

Exponential equations are mathematical statements where variables appear as exponents. This means you deal with equations like \( b^x = a \), where \( b \) is a constant base and \( x \) is the unknown variable. These types of equations often require us to find the exponent \( x \) that makes the equation true. To solve exponential equations, we sometimes express both sides of the equation with the same base, allowing us to equate the exponents directly. For instance, let’s say we have \( 4^x = 2 \). Realize that 2 can be rewritten as \( 4^{1/2} \) because the square root of 4 is 2. Now the equation \( 4^x = 4^{1/2} \) has the same base, and the exponents can be equated. This gives us \( x = 1/2 \). In summary, solving exponential equations often involves expressing numbers with the same base and simplifying the equation by comparing exponents.

Logarithmic Equations

Logarithmic equations use logarithms to relate two quantities. A basic logarithmic function is expressed as \( \log_b a = c \), meaning "find the power (\( c \)) to which the base (\( b \)) must be raised to yield \( a \)." Understanding these equations becomes easier when you recall that a logarithm is essentially the reverse of an exponent. Thus, every logarithmic equation can be rewritten as an exponential equation. For example, the equation \( \log_4 2 = x \) can be rewritten using the definition of the logarithm as \( 4^x = 2 \). Similarly, if given \( \log_4 x = 2 \), this means you need \( x \) such that when 4 is raised to the power of 2, it equals \( x \). Therefore, the exponential form, \( 4^2 = x \), directly shows that \( x = 16 \). This duality between logarithms and exponents is a powerful tool for solving logarithmic equations by transforming them into a form that’s often easier to solve.

Solving for x

Solving for \( x \) is commonly the goal in algebraic equations, whether dealing with exponential, logarithmic, or other types of equations. The key is to isolate \( x \) on one side of the equation, making it the subject. Let's consider two equations:

• \( \log_4 2 = x \)
• \( \log_4 x = 2 \)

For the first equation, we transform it into an exponential form as \( 4^x = 2 \), and determine that \( x = 1/2 \) by recognizing that 2 is the same as \( 4^{1/2} \). In the second equation, \( \log_4 x = 2 \), writing the exponential form gives \( 4^2 = x \), easily leading to \( x = 16 \). These transformations rely heavily on understanding the relationship between logarithmic and exponential forms. Once these forms are clear, solving for \( x \) becomes a matter of simplifying and rearranging the equations accordingly.