Problem 32 - 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. $$4 e^{7 x}=10,273$$

Step-by-Step Solution

Answer

The solution to the exponential equation is \(x = \frac{1}{7} ln\left(\frac{10273}{4}\right)\). The decimal approximation of the solution, accurate to two decimal places, will be the result obtained by plugging in \(\frac{1}{7} ln\left(\frac{10273}{4}\right)\) into a calculator.

Step 1: Isolate the Exponential Term

First, it's necessary to isolate the exponential term \(e^{7x}\) on one side of the equation. Do this by dividing both sides of the equation by 4. This gives: \[e^{7x} = \frac{10273}{4}\]

Step 2: Apply Natural Logarithm

Next, apply the natural logarithm (ln) to both sides in order to take the variable out of the exponent. The property of logarithms that says ln(a^b) = b*ln(a) can then be applied. This provides: \[ln(e^{7x}) = ln\left(\frac{10273}{4}\right)\] which simplifies to \[7x\cdot ln(e) = ln\left(\frac{10273}{4}\right)\] since ln(e) equals to 1, then it further simplifies to \[7x = ln\left(\frac{10273}{4}\right)\]

Step 3: Solve for x

Now, the task is to solve for 'x' by dividing both sides by 7. This provides: \[x = \frac{1}{7} ln\left(\frac{10273}{4}\right)\]

Step 4: Approximate Decimal Value

Finally, use a calculator to find the decimal approximation of 'x', rounded to two decimal places.