Problem 31 - 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. $$3 e^{5 x}=1977$$

Step-by-Step Solution

Answer

The solution to the equation \(3 e^{5 x}=1977\) is approximately \(x = \frac{\ln(\frac{1977}{3})}{5}\), where the precise numerical value would depend on the output from the calculator, computed to two decimal places.

Step 1: Isolate the exponential function

To begin, isolate the exponential function on one side of the equation. By dividing both sides by 3, the equation becomes: \(e^{5x} = \frac{1977}{3}\)

Step 2: Apply natural logarithm

Apply natural logarithm on both sides in order to get rid of the exponent. The inverse of the exponential function is the natural logarithm, which gives us: \(5x = \ln(\frac{1977}{3})\)

Step 3: Solve for x

Next, divide both sides of the equation by 5 to solve for \(x\): \(x = \frac{\ln(\frac{1977}{3})}{5}\)

Step 4: Numerical Approximation

Finally, use a calculator to evaluate the expression and round to two decimal places. Make sure that your calculator is set to the logarithm base e (natural logarithm).