Problem 252 - Calculus Volume 1

Question

For the following exercises, write the equation in equivalent exponential form. $$ \log _{9} 3=0.5 $$

Step-by-Step Solution

Answer

\( 9^{0.5} = 3 \)

Step 1: Understanding Logarithmic Form

The given logarithmic equation is \( \log_{9} 3 = 0.5 \). This means we need to find the exponential form that expresses the same relationship.

Step 2: Identifying Components

In logarithmic form, \( \log_{b} a = c \) can be interpreted as saying 'the power to which the base \( b \) must be raised to result in \( a \) is \( c \).' Here, \( b = 9 \), \( a = 3 \), and \( c = 0.5 \).

Step 3: Writing Exponential Form

In exponential form, the equation \( \log_{b} a = c \) becomes \( b^c = a \). Substitute the values: \( 9^{0.5} = 3 \). This is the desired equivalent exponential form.

Key Concepts

Logarithms

Logarithms are a way to express the concept of exponentiation in a reverse manner. When you see a logarithmic expression like \( \log_{b} a = c \), it simply asks the question, "To what power must we raise \( b \) to get \( a \)?" In this example, \( b \) is the base, \( a \) is the number we want to reach, and \( c \) represents the exponent or power. This means if you know \( b \) and \( a \), you can find \( c \). Logarithms help in understanding how quickly something grows or shrinks, like populations or investments.

• Logarithmic Form: The logarithmic form emphasizes finding an exponent within an equation.
• Example: Given \( \log_{9} 3 = 0.5 \), it tells us that 9 must be raised to the power of 0.5 to get 3.

Understanding how to read and convert logarithms is essential in many fields, especially those involving exponential growth and decay.

Base Conversion

Base conversion is often necessary when dealing with logarithms. It involves changing numbers from one base to another. When you're dealing with equations like \( \log_{b} a = c \), it's important to ensure you're working with a consistent base. For logarithms, understanding base conversion will enable you to interpret problems correctly and switch between different forms.

• Common Bases: Some of the most commonly used bases in mathematics are 10 (common log), \( e \) (natural log), and 2 (binary log).
• Formula: To convert a logarithm from one base to another, use the change of base formula: \( \log_{b} a = \frac{\log_{k} a}{\log_{k} b} \).
• Practical Application: This is especially helpful in computations involving different scales.

Base conversion ensures that you smoothly transition between different logarithmic forms. This flexibility is crucial for complex calculations.

Exponentiation

Exponentiation is the process of multiplying a base number by itself a certain number of times. It flips the script on logarithms, making it essential to derive exponential forms from logarithmic ones. In our exercise, transforming \( \log_{9} 3 = 0.5 \) into exponential form means finding what power 9 must be raised to become 3.

• Exponential Form: Here, \( b^c = a \) means raising \( b \) to the power of \( c \) gives us \( a \).
• Example: For \( 9^{0.5} = 3 \), 9 raised to the power of 0.5 equals 3, which is the square root of 9.
• Significance: Exponentiation helps in understanding growth processes, compound interest, and even natural phenomena.

Mastering exponentiation allows solving a wide array of mathematical problems, especially those involving large numbers and their behaviors.