Math  /  Algebra

QuestionFind the solution of the exponential equation 2ex9=72 e^{x}-9=7 in terms of logarithms, or correct to four decimal places. x=x= \square Next Question

Studdy Solution

STEP 1

What is this asking? We're asked to find the value of xx that makes 2ex9=72e^x - 9 = 7 true, and express it precisely using a logarithm, *or* approximately as a decimal. Watch out! Remember the order of operations!
We need to isolate exe^x *before* we can use logarithms.
Also, be careful with signs when rearranging the equation.

STEP 2

1. Isolate the exponential term
2. Apply the natural logarithm
3. Calculate the result

STEP 3

We want to get exe^x by itself.
The first thing we'll do is **add 9** to both sides of the equation to get rid of the 9-9 on the left.
This gives us: 2ex9+9=7+92e^x - 9 + 9 = 7 + 9 2ex=162e^x = 16Why did we add nine?
Because 9+9=0-9 + 9 = 0, so it removes the nine from the left side!

STEP 4

Now, we'll **divide** both sides by 2\textbf{2} to isolate exe^x: 2ex2=162 \frac{2e^x}{2} = \frac{16}{2} ex=8 e^x = 8 Since 22 divided by 22 is 11, we've successfully isolated exe^x!

STEP 5

To get xx by itself, we'll take the **natural logarithm** (ln) of both sides.
Remember, the natural logarithm is the inverse function of exe^x, so they "undo" each other! ln(ex)=ln(8) \ln(e^x) = \ln(8) x=ln(8) x = \ln(8) This works because ln(ex)\ln(e^x) simplifies to just xx!

STEP 6

Our exact solution is x=ln(8)x = \ln(8).
This is a perfectly valid answer!

STEP 7

If we want a decimal approximation, we can use a calculator to find ln(8)\ln(8). x2.0794 x \approx \textbf{2.0794} We've rounded to four decimal places, as requested!

STEP 8

The exact solution is x=ln(8)x = \ln(8), and the approximate solution is x2.0794x \approx 2.0794.

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