Problem 72 - Algebra 2

Question

Write an equivalent exponential equation. $$ \log _{3} x=2 $$

Step-by-Step Solution

Answer

The equivalent exponential equation is \( 3^2 = x \).

Step 1: Recall Logarithm Definition

To rewrite the given logarithmic equation as an exponential equation, we must first recall the definition of a logarithm. Specifically, the equation \( \log_{b}a = c \) is equivalent to the exponential form \( b^c = a \).

Step 2: Identify Components of the Logarithmic Equation

In the given equation \( \log_{3} x = 2 \), identify the base \( b = 3 \), the argument \( a = x \), and the logarithmic result \( c = 2 \).

Step 3: Convert to Exponential Form

Using the components identified, apply the logarithm definition to rewrite the equation in exponential form. Substitute \( b = 3 \), \( c = 2 \), and \( a = x \) into the format of \( b^c = a \).

Step 4: Write the Equivalent Exponential Equation

Based on the conversion, the equivalent exponential equation is \( 3^2 = x \).

Key Concepts

Logarithmic Equations

Logarithmic equations are mathematical statements that relate logarithms with their corresponding values. In a logarithmic equation, such as \( \log_{b} a = c \), there is a relationship between three components: the base \( b \), the argument \( a \), and the result \( c \). Here, the logarithm "\( \log_{b} a \)" signifies the power to which the base \( b \) must be raised to equal the argument \( a \).
Some common characteristics of logarithmic equations include:

• They are used to undo exponential functions, much like how division is the inverse operation of multiplication.
• They often appear in problems dealing with exponential growth or decay, such as in finance or science.

This concept becomes particularly useful when solving equations where the unknown variable is an exponent.

Conversion of Logarithms to Exponentials

Conversion from a logarithmic to an exponential equation is a fundamental technique in understanding and solving logarithmic equations. It utilizes the inherent definition of logarithms, helping to bridge the gap between logarithmic and exponential forms. This is particularly useful when solving equations that can be more challenging to understand in their logarithmic form.
To convert a logarithm to an exponential equation, you can follow these steps:

• Identify the base \( b \) of the logarithm, the argument \( a \), and the result \( c \).
• Use the relation \( \log_{b} a = c \) which is equivalent to \( b^c = a \).

Applying this technique to our example, the equation \( \log_{3} x = 2 \) converts to an exponential form as \( 3^2 = x \), making it easier to solve.

Properties of Logarithms

The properties of logarithms are tools that allow simplification and manipulation of logarithmic expressions. Understanding these properties can make solving logarithmic equations more intuitive and provide shortcuts in otherwise complex calculations.

• Product Property: \( \log_{b}(mn) = \log_{b}m + \log_{b}n \)
• Quotient Property: \( \log_{b}(m/n) = \log_{b}m - \log_{b}n \)
• Power Property: \( \log_{b}(m^n) = n \cdot \log_{b}m \)
• Base Change Property: Converts between bases: \( \log_{b}a = \frac{\log_{k}a}{\log_{k}b} \)

These properties help transform complex logarithmic expressions into more manageable forms, assisting in the conversion between logarithmic and exponential forms.

Definition of Logarithms

Understanding the definition of logarithms is crucial when working with these equations. The logarithm \( \log_{b} a \) is defined as the power to which the base \( b \) must be raised to produce \( a \). This definition is pivotal because it infers that every logarithmic expression has a corresponding exponential form.
Consider the expression \( \log_{3} x = 2 \). Here, we are finding the power to which 3 must be raised to give the result \( x \). By the definition, this expression translates into the exponential equation \( 3^2 = x \).
Logarithms help address problems involving growth, decay, and other phenomena where exponential change is observed, providing a strong mathematical foundation for real-world applications.