Problem 42 - 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. $$7^{2 x+1}=3^{x+2}$$

Step-by-Step Solution

Answer

The solution to the equation in terms of natural logarithms is \(x = \frac{2\ln(3) - \ln(7)}{2\ln(7) - \ln(3)}\), and the decimal approximation for x is approximately -0.87.

Step 1: Apply natural logarithm

The first step is to take the natural logarithm of both sides. So our equation \(7^{2x+1} = 3^{x+2}\) transforms into \(\ln(7^{2x+1}) = \ln(3^{x+2}).\)

Step 2: Use logarithm properties

We can now use the property of logs that allows us to bring the exponent down in front of the log. This changes our equation into \((2x+1)\ln(7) = (x+2)\ln(3)\).

Step 3: Distribute and arrange

Next, distribute the multiplication and arrange terms: \(2x\ln(7) + \ln(7) = x\ln(3) + 2\ln(3)\). Move the terms with x to one side and the others to the other, getting \(2x\ln(7) - x\ln(3) = 2\ln(3) - \ln(7)\).

Step 4: Factor out x

Now, factor out x from the left side of the equation: \(x[2\ln(7) - \ln(3)] = 2\ln(3) - \ln(7)\).

Step 5: Solve for x

Lastly, solve for x by dividing both sides by \([2\ln(7) - \ln(3)]\). Then, it leaves us with \(x = \frac{2\ln(3) - \ln(7)}{2\ln(7) - \ln(3)}\).

Step 6: Evaluate to two decimal places

Using a calculator, we can evaluate, to two decimal places, the value for x, which is around -0.87.