Problem 47 - College Algebra Essentials

Question

Solve each exponential equation. 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. $$3^{2 x}+3^{x}-2=0$$

Step-by-Step Solution

Answer

The solution to the equation \(3^{2x}+3^{x}-2=0\) is \(x = 0\), as \(3^x\) can't be negative and \( log_3(1) = 0\).

Step 1: Recognize as Quadratic Form

The equation given is similar to a quadratic equation, if we look at it with a substitution. Let's take \(y = 3^x\), so our equation now becomes \(y^2 + y - 2 = 0\). We have just converted the equation into a quadratic formula, which makes it easier to find solutions for \(y\).

Step 2: Solve the Quadratic Equation

Now, solve this quadratic equation by using the quadratic formula \(y = [-b ± sqrt(b^2-4ac)] / (2a)\). Substitution these values: \(a=1\), \(b=1\) and \(c=-2\), into the formula gives us \(y=[-1 ± sqrt((1)^2-4*1*(-2))] / 2(1)\), simplifying further gets \(y=-1 ± sqrt(9) / 2\). Therefore, the solutions for \(y\) are \(y1 = -2\) and \(y2 = 1\).

Step 3: Back-substitution and Logarithmic Conversion

Remember we made a substitution in Step 1, so now we replace \(y\) with \(3^x\). Setting this equal to our solutions gives us two new equations: \(3^x = -2 \) and \(3^x = 1\). However, \(3^x\) can't be negative so we discard the \(3^x = -2\) solution. Solving for \(x\) in the latter equation by taking the natural logarithm of both sides, we get \(x = log_3(1)\). Logarithm of 1 in any base is equal to zero, so we have \(x = 0\).

Step 4: Decimal Approximation

Lastly, the decimal approximation for the solution should be calculated by using a calculator. However, in this case, since the solution is \(x = 0\), there's no further need for a decimal approximation.