Problem 70 - Algebra 2

Question

Give an example of an exponential equation and its related logarithmic equation.

Step-by-Step Solution

Answer

Example: Exponential equation: \( 2^x = 8 \); Logarithmic equation: \( \log_2(8) = x \). Solution: \( x = 3 \).

Step 1: Understanding Exponential Equations

An exponential equation is a mathematical statement where an unknown variable appears as an exponent. A common form looks like \( a^x = b \), where \( a \) is a constant, \( x \) is the exponent or variable, and \( b \) is a constant as well. We will use an example to demonstrate.

Step 2: Choose an Example Exponential Equation

Let's select the exponential equation \( 2^x = 8 \). In this equation, the base is 2, the exponent is \( x \), and the output or result is 8.

Step 3: Rewriting as a Logarithmic Equation

To convert from an exponential equation \( 2^x = 8 \) to a logarithmic form, use the definition of a logarithm: \( \log_a(b) = x \) means \( a^x = b \). Thus, the exponential equation \( 2^x = 8 \) becomes \( \log_2(8) = x \).

Step 4: Solving the Logarithmic Equation

The logarithmic equation \( \log_2(8) = x \) can be evaluated by recognizing that 8 is 2 raised to the power of 3. Hence, \( x = 3 \), as \( 2^3 = 8 \).

Key Concepts

Logarithmic Equations

Logarithmic equations are fascinating mathematical statements that involve the concept of logarithms. These equations express the relationship between exponents and their resulting values in a way that is easy to interpret and solve. When we deal with a logarithmic equation, it is often written in the form \( \log_a(b) = x \), where:

• \( a \) is the base of the log, similar to the base in an exponential form.
• \( b \) is the value we are taking the log of.
• \( x \) is the exponent, representing the unknown quantity we often seek to find.

Logarithmic equations effectively transpose exponential equations to a form that is often more conducive to solving, especially when the input values are not power of the base. By converting an exponential equation like \( 2^x = 8 \) into the logarithmic form \( \log_2(8) = x \), it becomes easier to deduce that \( x = 3 \) because 8 is actually \( 2^3 \). This transformation highlights how logarithmic equations serve as an essential tool for working with exponential relationships.

Exponents

Exponents are powerful tools in mathematics that allow us to express repeated multiplication in a concise form. An exponent indicates how many times a base number is multiplied by itself. For instance, in the expression \( a^x \):

• \( a \) is the base, the number being multiplied.
• \( x \) is the exponent, telling us how many times to multiply the base by itself.

Understanding exponents is crucial when working with exponential equations, such as \( 2^x = 8 \). Here, the base 2 is used, with \( x \) representing the unknown number of times it is multiplied to obtain 8. This translates to solving for \( x \) when 2 is raised to the power of \( x \) to produce 8. Recognizing patterns in exponents helps simplify complex calculations and solve equations by examining powers of numbers.

Logarithms

Logarithms are the inverse operations of exponents. If exponents are all about expanding numbers by multiplying them by themselves, logarithms are about determining how many times you must multiply a base to reach a certain number. The notation \( \log_a(b) \) reads as "the logarithm of \( b \), base \( a \)." It solves the mystery: "To what power must I raise \( a \) to get \( b \)?"

Logarithms streamline certain types of calculations by converting multiplication problems into simpler addition problems. For example, understanding that \( 2^x = 8 \) in exponential form, we translate this to \( \log_2(8) = x \) in logarithmic form. Recognizing the power pattern—since \( 2^3 = 8 \)—logarithms confirm \( x = 3 \). This functionality makes logarithms indispensable tools for solving a wide range of mathematical problems.

Mathematical Statements

Mathematical statements are expressions that present a factual claim, often involving variables and constants, which can be either true or false. In the realm of solving equations, these statements are fundamental as they define what is being asked or solved, such as in exponential or logarithmic contexts.

• An exponential statement like \( 2^x = 8 \) suggests a process for finding \( x \) by understanding powers of numbers.
• The transition to a logarithmic statement \( \log_2(8) = x \) provides an alternative method to express and solve for \( x \).

These mathematical statements give structure to calculations and problem solving. They help clarify the relationships between variables and constants by framing them in accessible and logical formats. Ultimately, they serve as the foundation to solve real-world problems, educate learners about mathematical principles, and foster a deeper comprehension of numerical concepts.