Problem 29 - Algebra 2

Question

Write an equivalent exponential or logarithmic equation. $$ \ln 4+2 \ln x=8 $$

Step-by-Step Solution

Answer

The equivalent equation is \( 4x^2 = e^8 \).

Step 1 - Apply Logarithmic Property

Use the property of logarithms that states: \( a \ln b = \ln(b^a) \). According to this property, \( 2 \ln x \) can be rewritten as \( \ln(x^2) \). So the equation \( \ln 4 + 2 \ln x = 8 \) becomes \( \ln 4 + \ln x^2 = 8 \).

Step 2 - Combine Logarithms

Combine the two logarithms on the left side using the logarithmic property: \( \ln a + \ln b = \ln(ab) \). Therefore, \( \ln 4 + \ln x^2 \) becomes \( \ln(4x^2) \). Substitute this back into the equation to get \( \ln(4x^2) = 8 \).

Step 3 - Convert to Exponential Form

Convert the logarithmic equation \( \ln(4x^2) = 8 \) to its exponential form using the property: \( \ln a = b \) is equivalent to \( a = e^b \). Therefore, \( 4x^2 = e^8 \) is the equivalent exponential equation to the original problem.

Key Concepts

Understanding Logarithmic Properties

Logarithms are mathematical expressions that help quantify how many times we multiply a number by itself to get another given number. In simpler terms, if you know the result of an exponentiation and you need to find the exponent, you'll use a logarithm. One of the key properties of logarithms used in the exercise is the power property, which states that multiplying a logarithm by an exponent is equivalent to taking the logarithm of that number raised to the power of the exponent. This can be expressed as:

• \( a \ln b = \ln(b^a) \)

This property is helpful because it allows us to simplify expressions where a logarithm is multiplied by a constant. For example, if you have \( 2 \ln x \), it can be rewritten using the power property as \( \ln(x^2) \). This makes it easier to combine and manipulate logarithmic expressions. Always remember that understanding and applying properties like the power property can transform complex expressions into simpler, more workable forms.

Working with Exponential Equations

Exponential equations are equations where variables appear as exponents. These equations often describe rapid growth or decay processes and are fundamental in various fields like finance, science, and engineering. In our solution, the equation \( \ln(4x^2) = 8 \) is transformed using our understanding of logarithms into an exponential equation. To convert a logarithmic equation to its exponential form, we rely on the relationship:

• \( \ln(a) = b \) translates to \( a = e^b \)

In this context, \( \ln(4x^2) = 8 \) becomes \( 4x^2 = e^8 \). Recognizing and applying this conversion is crucial because it allows us to work directly with the exponential components, which can be more intuitive in solving problems that involve growth or multiplying factors. Exponential equations often reveal the underlying structure in problems involving repeated multiplication processes.

Exploring Equivalent Equations

Equivalent equations are different expressions that represent the same relationships or values. They are a fundamental concept in algebra, where manipulating an equation through valid mathematical operations results in a new expression that's equal in value to the original. In the exercise solution, equivalent equations are demonstrated by converting between logarithmic and exponential forms. By understanding the relationships and properties that link these forms, you can confidently transform equations between different states while maintaining their equivalence. For instance, transforming \( \ln(4x^2) = 8 \) to \( 4x^2 = e^8 \) shows how you can switch between forms based on the context or solving needs.Creating equivalent equations can simplify problem-solving, make it more intuitive, and reveal new insights or solutions to the problem at hand. Mastery of this concept allows for greater flexibility and creativity in mathematical problem-solving, as it gives multiple paths towards finding solutions or expressions.