Solved on Dec 18, 2023

Solve for xx in the equation ln(x2)3=9\frac{\ln (x-2)}{3}=9.

STEP 1

Assumptions
1. We are solving the equation for xx.
2. We assume that x>2x > 2 since the natural logarithm function, ln(x)\ln(x), is defined only for positive real numbers and x2x-2 must be positive.
3. We will use properties of logarithms and exponentiation to solve the equation.

STEP 2

First, we need to isolate the natural logarithm function by multiplying both sides of the equation by 3.
3ln(x2)3=393 \cdot \frac{\ln (x-2)}{3} = 3 \cdot 9

STEP 3

Simplify both sides of the equation.
ln(x2)=27\ln (x-2) = 27

STEP 4

Now, we will exponentiate both sides of the equation to remove the natural logarithm. We use the fact that eln(a)=ae^{\ln(a)} = a for any positive number aa.
eln(x2)=e27e^{\ln (x-2)} = e^{27}

STEP 5

Simplify the left side of the equation using the property of logarithms mentioned above.
x2=e27x - 2 = e^{27}

STEP 6

Next, we will solve for xx by adding 2 to both sides of the equation.
x=e27+2x = e^{27} + 2

STEP 7

Now we have the solution for xx. However, e27e^{27} is a very large number, and it is typically left in the exponential form unless a decimal approximation is required.
The solution for the equation is:
x=e27+2x = e^{27} + 2

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