Problem 32 - Algebra and Trigonometry

Question

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 = 1/2; (b) x = 16.

Step 1: Understand the Definition of Logarithm

The logarithm \( \log_b a = c \) is defined such that \( b^c = a \). It states that \( b \) raised to the power of \( c \) equals \( a \). We'll apply this to solve for \( x \).

Step 2: Solve Part (a) \( \log_4 2 = x \)

Using the logarithm definition, set \( 4^x = 2 \). This equation means the base 4 raised to the power of \( x \) will equal 2. To solve for \( x \), we need to find a power of 4 that results in 2. Recognizing 2 as an adjustment of \( 4^{1/2} \) (since \( 2 = \sqrt{4} \)), we find \( x = \frac{1}{2} \).

Step 3: Solve Part (b) \( \log_4 x = 2 \)

Set the equation according to the definition: \( 4^2 = x \). Calculate \( 4^2 \) which equals 16, thus \( x = 16 \).

Key Concepts

Definition of Logarithms

Understanding the definition of logarithms is crucial for solving equations that involve them. A logarithm answers the question: "To what power must the base be raised, to produce a given number?" For instance, in the expression \( \log_b a = c \),

• \( b \) is the base of the logarithm
• \( a \) is the number you want to find the logarithm for
• \( c \) is the exponent or power that the base must be raised to, in order to equal \( a \)

So, log\( _b \) \( a \) = \( c \) means \( b^c = a \). This is the fundamental relationship that defines logarithms. It's a powerful tool that lets us transform multiplicative situations into additive ones, simplifying complex mathematical problems. Think of it as the inverse operation of exponentiation.

Solving Logarithmic Equations

Solving logarithmic equations involves applying the definition of logarithms to rewrite the log equation as an exponential equation. For example, if you have \( \log_4 2 = x \), you rewrite it using the fact that it means \( 4^x = 2 \).

• You determine that \( 4^x \) should equal 2.
• Knowing that \( 2 = 4^{1/2} \) since 2 is the square root of 4, helps you see that \( x = \frac{1}{2} \).

For another case, such as \( \log_4 x = 2 \), the definition yields \( 4^2 = x \), leading directly to the fact that \( x = 16 \). The procedure effectively finds ‘\( x \)’ by determining what power (in the first case) or result (in the second case) aligns with the logarithmic definition when applied to a specific base.

Properties of Exponents

Exponents are closely linked with logarithms and share fundamental properties that are very useful when handling logarithmic equations.

• Multiplication: \( a^m \times a^n = a^{m+n} \)
• Division: \( \frac{a^m}{a^n} = a^{m-n} \)
• Power to a Power: \((a^m)^n = a^{m \times n} \)
• Root: \( a^{1/n} = \sqrt[n]{a} \)

These properties help you manipulate expressions when solving logarithmic equations. For example, knowing how to express roots as fractional exponents allows you to identify that \( 4^{1/2} = 2 \), which can simplify finding solutions to equations involving logs. Recognizing these connections is key to understanding and working effectively with exponential and logarithmic expressions.