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 to clearly identify the exponential terms: $3^{2x}+3^x-\; 2\; =\; 0$.
2. Recognize that $3^{2x}={3^x}^2$, so let $y\; =\; 3^x$ to simplify the equation.
3. Substitute $y\; =\; 3^x$ into the equation to get a quadratic form: $y^2\; +\; y\; -\; 2\; =\; 0$.
4. Solve the quadratic equation $y^2\; +\; y\; -\; 2\; =\; 0$ using the quadratic formula: $y\; =\; \backslash frac\{-b\; \backslash pm\; \backslash sqrt\{b^2\; -\; 4ac\}\}\{2a\}$, where $a=1$, $b=1$, and $c=-2$.
5. After finding the values of $y$, substitute back $y\; =\; 3^x$ and solve for $x$ by taking the natural logarithm or common logarithm: $x\; =\; \backslash frac\{\backslash ln(y)\}\{\backslash ln(3)\}$ or $x\; =\; \backslash frac\{\backslash log(y)\}\{\backslash log(3)\}$.

Key Concepts

Exponential Equations

Exponential equations involve variables in the exponent, such as 3^(2x) or 3^x. Solving these requires rewriting terms to have the same base or using substitution to simplify the equation before isolating the variable.

Logarithms and Their Properties

Logarithms are the inverse operations of exponentials and help solve equations where the variable is an exponent. Natural logarithms (ln) and common logarithms (log) allow us to rewrite exponential equations into linear forms to isolate the variable.

Using a Calculator for Approximation

After expressing the solution in logarithmic form, calculators are used to find decimal approximations. This step involves evaluating logarithmic expressions and rounding the result to the desired decimal places, ensuring practical and interpretable answers.