Solve each exponential equation in Exercises 23–48. Express the solution set in terms of natural logarithms or common logarithms. Then use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. 3^2x+3^x−2=0

Verified step by step guidance

1. Rewrite the equation in terms of a single variable by letting \( y = 3^x \). This transforms the equation into \( y^2 + y - 2 = 0 \).
2. Recognize that the equation \( y^2 + y - 2 = 0 \) is a quadratic equation. Use the quadratic formula \( y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) to solve for \( y \), where \( a = 1 \), \( b = 1 \), and \( c = -2 \).
3. Calculate the discriminant \( b^2 - 4ac \) to determine the nature of the roots.
4. Solve for \( y \) using the quadratic formula, which will give you two potential solutions for \( y \).
5. Since \( y = 3^x \), solve for \( x \) by taking the logarithm of both sides for each solution of \( y \). Use either natural logarithms (\( \ln \)) or common logarithms (\( \log \)) to express \( x \) in terms of logarithms.

Key Concepts

Exponential Equations

Exponential equations are equations in which variables appear as exponents. To solve these equations, one often needs to manipulate the equation to isolate the exponential term, allowing for the application of logarithms. Understanding the properties of exponents, such as the product, quotient, and power rules, is essential for simplifying and solving these types of equations.

Logarithms

Logarithms are the inverse operations of exponentiation, allowing us to solve for the exponent in an exponential equation. There are two common types of logarithms: natural logarithms (base e) and common logarithms (base 10). Knowing how to convert between exponential and logarithmic forms is crucial for solving exponential equations, as it enables us to express solutions in a more manageable form.

Calculator Use for Approximations

Using a calculator to obtain decimal approximations is an important step in solving exponential equations. After expressing the solution in terms of logarithms, a calculator can provide numerical values that are often necessary for practical applications. Understanding how to input logarithmic expressions into a calculator and round results to a specified number of decimal places is essential for accuracy in solutions.