Solved on Dec 18, 2023

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

STEP 1

Assumptions
1. We are solving the equation for $x$ .
2. We assume that $x > 2$ since the natural logarithm function, $\ln(x)$ , is defined only for positive real numbers and $x-2$ 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.
$3 \cdot \frac{\ln (x-2)}{3} = 3 \cdot 9$

STEP 3

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

STEP 4

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

STEP 5

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

STEP 6

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

STEP 7

Now we have the solution for $x$ . However, $e^{27}$ 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:
$x = e^{27} + 2$

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

STEP 1

Assumptions
1. We are solving the equation for $x$ .
2. We assume that $x > 2$ since the natural logarithm function, $\ln(x)$ , is defined only for positive real numbers and $x-2$ 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.
$3 \cdot \frac{\ln (x-2)}{3} = 3 \cdot 9$

STEP 3

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

STEP 4

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

STEP 5

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

STEP 6

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

STEP 7

Now we have the solution for $x$ . However, $e^{27}$ 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:
$x = e^{27} + 2$

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Solve for xxx in the equation ln⁡(x−2)3=9\frac{\ln (x-2)}{3}=93ln(x−2)​=9.

STEP 1

Assumptions1. We are solving the equation for xxx.2. We assume that x>2x > 2x>2 since the natural logarithm function, ln⁡(x)\ln(x)ln(x), is defined only for positive real numbers and x−2x-2x−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 3.3⋅ln⁡(x−2)3=3⋅93 \cdot \frac{\ln (x-2)}{3} = 3 \cdot 93⋅3ln(x−2)​=3⋅9

STEP 3

Simplify both sides of the equation.ln⁡(x−2)=27\ln (x-2) = 27ln(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)} = aeln(a)=a for any positive number aaa.eln⁡(x−2)=e27e^{\ln (x-2)} = e^{27}eln(x−2)=e27

STEP 5

Simplify the left side of the equation using the property of logarithms mentioned above.x−2=e27x - 2 = e^{27}x−2=e27

STEP 6

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

STEP 7

Now we have the solution for xxx. However, e27e^{27}e27 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} + 2x=e27+2

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Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

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

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

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

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

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

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

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

Now we have the solution for $x$ . However, $e^{27}$ 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:
$x = e^{27} + 2$