Math  /  Algebra

QuestionEvaluate y=ln(x2)y=\ln (x-2) for the following values of xx. Round to the nearest thousandth. x=3,y=x=3, y= \square x=4,yx=4, y \approx \square x=6,yx=6, y \approx \square

Studdy Solution

STEP 1

What is this asking? We need to plug different values of xx into the equation y=ln(x2)y = \ln(x-2) and calculate the corresponding values of yy, rounded to the nearest thousandth. Watch out! Remember that the natural logarithm ln\ln is only defined for positive inputs.
So, x2x-2 must be greater than zero!

STEP 2

1. Evaluate for x=3x=3
2. Evaluate for x=4x=4
3. Evaluate for x=6x=6

STEP 3

Let's **substitute** x=3x = 3 into our equation: y=ln(x2)y = \ln(x-2).
This gives us y=ln(32)y = \ln(3-2).

STEP 4

Inside the logarithm, we have 32=13-2=1, so our equation becomes y=ln(1)y = \ln(1).

STEP 5

Now, we remember that the natural logarithm of **1** is always **0**.
So, y=ln(1)=0y = \ln(1) = 0.
That was easy!

STEP 6

Now, let's **plug in** x=4x = 4 into y=ln(x2)y = \ln(x-2).
This gives us y=ln(42)y = \ln(4-2).

STEP 7

Simplifying inside the logarithm, we get 42=24-2=2, so y=ln(2)y = \ln(2).

STEP 8

Using a calculator, we find that ln(2)0.693147\ln(2) \approx 0.693147.
We need to round this to the nearest thousandth, which gives us **0.693**.

STEP 9

Let's **substitute** x=6x = 6 into y=ln(x2)y = \ln(x-2), giving us y=ln(62)y = \ln(6-2).

STEP 10

Inside the logarithm, 62=46-2=4, so we have y=ln(4)y = \ln(4).

STEP 11

Using a calculator, we find ln(4)1.386294\ln(4) \approx 1.386294.
Rounding to the nearest thousandth gives us **1.386**.

STEP 12

For x=3x=3, y=0y=0. For x=4x=4, y0.693y \approx 0.693. For x=6x=6, y1.386y \approx 1.386.

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