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. Start by recognizing that the equation involves exponential expressions with the same base: \(3^{2x} + 3^x - 2 = 0\). Notice that \$3^{2x}\( can be rewritten as \)(3^x)^2$.
2. Introduce a substitution to simplify the equation. Let \(y = 3^x\). Then the equation becomes \(y^2 + y - 2 = 0\).
3. Solve the quadratic equation \(y^2 + y - 2 = 0\) using the quadratic formula: \(y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a=1\), \(b=1\), and \(c=-2\).
4. After finding the values of \(y\), recall that \(y = 3^x\). Solve for \(x\) by taking the logarithm of both sides: \(x = \log_3(y)\). You can express this using natural logarithms as \(x = \frac{\ln(y)}{\ln(3)}\) or common logarithms as \(x = \frac{\log(y)}{\log(3)}\).
5. Evaluate the logarithmic expressions using a calculator to find the decimal approximations of \(x\), rounding to two decimal places as required.

Key Concepts

Exponential Equations

Exponential equations involve variables in the exponent, such as 3^(2x) or 3^x. Solving these requires rewriting the equation to isolate the exponential expressions or to express them in a common base, enabling the use of algebraic methods to find the variable.

Logarithms and Their Properties

Logarithms are the inverse operations of exponentials, allowing us to solve for variables in exponents. Natural logarithms (ln) and common logarithms (log) help convert exponential equations into linear forms, making it easier to isolate and solve for the unknown.

Quadratic Form in Exponential Equations

Some exponential equations can be transformed into quadratic equations by substituting expressions like 3^x = y. This substitution simplifies the problem to solving a quadratic equation, after which the original variable can be found by reversing the substitution.