Solved on Dec 15, 2023

Solve the logarithmic equation $\ln (7 x-3)-5=-3$ for the value of $x$ .

STEP 1

Assumptions
1. We are given the equation $\ln(7x - 3) - 5 = -3$ .
2. We need to solve for $x$ .
3. We assume that $x$ is in the domain of the natural logarithm function, which means $7x - 3 > 0$ .

STEP 2

To isolate the logarithmic term, we need to add 5 to both sides of the equation.
$\ln(7x - 3) - 5 + 5 = -3 + 5$

STEP 3

Simplify the equation by combining like terms.
$\ln(7x - 3) = 2$

STEP 4

To solve for $x$ , we need to eliminate the natural logarithm. We do this by raising $e$ to the power of both sides of the equation, since $e^{\ln(a)} = a$ .
$e^{\ln(7x - 3)} = e^2$

STEP 5

Apply the property of logarithms that states $e^{\ln(a)} = a$ .
$7x - 3 = e^2$

STEP 6

Add 3 to both sides of the equation to isolate the term with $x$ .
$7x - 3 + 3 = e^2 + 3$

STEP 7

Simplify the equation by combining like terms.
$7x = e^2 + 3$

STEP 8

To solve for $x$ , divide both sides of the equation by 7.
$x = \frac{e^2 + 3}{7}$

STEP 9

Calculate the value of $e^2$ using a calculator or known value.
$e^2 \approx 7.38905609893$

STEP 10

Substitute the value of $e^2$ into the equation.
$x = \frac{7.38905609893 + 3}{7}$

STEP 11

Calculate the numerator by adding 3 to the value of $e^2$ .
$x = \frac{10.38905609893}{7}$

STEP 12

Divide the numerator by 7 to find the value of $x$ .
$x \approx \frac{10.38905609893}{7} \approx 1.48415158556$
So, the solution to the equation $\ln(7x - 3) - 5 = -3$ is approximately $x \approx 1.48415158556$ .

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Solve the logarithmic equation $\ln (7 x-3)-5=-3$ for the value of $x$ .

STEP 1

Assumptions
1. We are given the equation $\ln(7x - 3) - 5 = -3$ .
2. We need to solve for $x$ .
3. We assume that $x$ is in the domain of the natural logarithm function, which means $7x - 3 > 0$ .

STEP 2

To isolate the logarithmic term, we need to add 5 to both sides of the equation.
$\ln(7x - 3) - 5 + 5 = -3 + 5$

STEP 3

Simplify the equation by combining like terms.
$\ln(7x - 3) = 2$

STEP 4

To solve for $x$ , we need to eliminate the natural logarithm. We do this by raising $e$ to the power of both sides of the equation, since $e^{\ln(a)} = a$ .
$e^{\ln(7x - 3)} = e^2$

STEP 5

Apply the property of logarithms that states $e^{\ln(a)} = a$ .
$7x - 3 = e^2$

STEP 6

Add 3 to both sides of the equation to isolate the term with $x$ .
$7x - 3 + 3 = e^2 + 3$

STEP 7

Simplify the equation by combining like terms.
$7x = e^2 + 3$

STEP 8

To solve for $x$ , divide both sides of the equation by 7.
$x = \frac{e^2 + 3}{7}$

STEP 9

Calculate the value of $e^2$ using a calculator or known value.
$e^2 \approx 7.38905609893$

STEP 10

Substitute the value of $e^2$ into the equation.
$x = \frac{7.38905609893 + 3}{7}$

STEP 11

Calculate the numerator by adding 3 to the value of $e^2$ .
$x = \frac{10.38905609893}{7}$

STEP 12

Divide the numerator by 7 to find the value of $x$ .
$x \approx \frac{10.38905609893}{7} \approx 1.48415158556$
So, the solution to the equation $\ln(7x - 3) - 5 = -3$ is approximately $x \approx 1.48415158556$ .

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Solve the logarithmic equation ln⁡(7x−3)−5=−3\ln (7 x-3)-5=-3ln(7x−3)−5=−3 for the value of xxx.

STEP 1

Assumptions1. We are given the equation ln⁡(7x−3)−5=−3\ln(7x - 3) - 5 = -3ln(7x−3)−5=−3.2. We need to solve for xxx.3. We assume that xxx is in the domain of the natural logarithm function, which means 7x−3>07x - 3 > 07x−3>0.

STEP 2

To isolate the logarithmic term, we need to add 5 to both sides of the equation.ln⁡(7x−3)−5+5=−3+5\ln(7x - 3) - 5 + 5 = -3 + 5ln(7x−3)−5+5=−3+5

STEP 3

Simplify the equation by combining like terms.ln⁡(7x−3)=2\ln(7x - 3) = 2ln(7x−3)=2

STEP 4

To solve for xxx, we need to eliminate the natural logarithm. We do this by raising eee to the power of both sides of the equation, since eln⁡(a)=ae^{\ln(a)} = aeln(a)=a.eln⁡(7x−3)=e2e^{\ln(7x - 3)} = e^2eln(7x−3)=e2

STEP 5

Apply the property of logarithms that states eln⁡(a)=ae^{\ln(a)} = aeln(a)=a.7x−3=e27x - 3 = e^27x−3=e2

STEP 6

Add 3 to both sides of the equation to isolate the term with xxx.7x−3+3=e2+37x - 3 + 3 = e^2 + 37x−3+3=e2+3

STEP 7

Simplify the equation by combining like terms.7x=e2+37x = e^2 + 37x=e2+3

STEP 8

To solve for xxx, divide both sides of the equation by 7.x=e2+37x = \frac{e^2 + 3}{7}x=7e2+3​

STEP 9

Calculate the value of e2e^2e2 using a calculator or known value.e2≈7.38905609893e^2 \approx 7.38905609893e2≈7.38905609893

STEP 10

Substitute the value of e2e^2e2 into the equation.x=7.38905609893+37x = \frac{7.38905609893 + 3}{7}x=77.38905609893+3​

STEP 11

Calculate the numerator by adding 3 to the value of e2e^2e2.x=10.389056098937x = \frac{10.38905609893}{7}x=710.38905609893​

STEP 12

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Solve the logarithmic equation $\ln (7 x-3)-5=-3$ for the value of $x$ .

Assumptions
1. We are given the equation $\ln(7x - 3) - 5 = -3$ .
2. We need to solve for $x$ .
3. We assume that $x$ is in the domain of the natural logarithm function, which means $7x - 3 > 0$ .

To isolate the logarithmic term, we need to add 5 to both sides of the equation.
$\ln(7x - 3) - 5 + 5 = -3 + 5$

Simplify the equation by combining like terms.
$\ln(7x - 3) = 2$

To solve for $x$ , we need to eliminate the natural logarithm. We do this by raising $e$ to the power of both sides of the equation, since $e^{\ln(a)} = a$ .
$e^{\ln(7x - 3)} = e^2$

Apply the property of logarithms that states $e^{\ln(a)} = a$ .
$7x - 3 = e^2$

Add 3 to both sides of the equation to isolate the term with $x$ .
$7x - 3 + 3 = e^2 + 3$

Simplify the equation by combining like terms.
$7x = e^2 + 3$

To solve for $x$ , divide both sides of the equation by 7.
$x = \frac{e^2 + 3}{7}$

Calculate the value of $e^2$ using a calculator or known value.
$e^2 \approx 7.38905609893$

Substitute the value of $e^2$ into the equation.
$x = \frac{7.38905609893 + 3}{7}$

Calculate the numerator by adding 3 to the value of $e^2$ .
$x = \frac{10.38905609893}{7}$