Solved on Dec 15, 2023

Solve the logarithmic equation ln(7x3)5=3\ln (7 x-3)-5=-3 for the value of xx.

STEP 1

Assumptions
1. We are given the equation ln(7x3)5=3\ln(7x - 3) - 5 = -3.
2. We need to solve for xx.
3. We assume that xx is in the domain of the natural logarithm function, which means 7x3>07x - 3 > 0.

STEP 2

To isolate the logarithmic term, we need to add 5 to both sides of the equation.
ln(7x3)5+5=3+5\ln(7x - 3) - 5 + 5 = -3 + 5

STEP 3

Simplify the equation by combining like terms.
ln(7x3)=2\ln(7x - 3) = 2

STEP 4

To solve for xx, we need to eliminate the natural logarithm. We do this by raising ee to the power of both sides of the equation, since eln(a)=ae^{\ln(a)} = a.
eln(7x3)=e2e^{\ln(7x - 3)} = e^2

STEP 5

Apply the property of logarithms that states eln(a)=ae^{\ln(a)} = a.
7x3=e27x - 3 = e^2

STEP 6

Add 3 to both sides of the equation to isolate the term with xx.
7x3+3=e2+37x - 3 + 3 = e^2 + 3

STEP 7

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

STEP 8

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

STEP 9

Calculate the value of e2e^2 using a calculator or known value.
e27.38905609893e^2 \approx 7.38905609893

STEP 10

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

STEP 11

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

STEP 12

Divide the numerator by 7 to find the value of xx.
x10.3890560989371.48415158556x \approx \frac{10.38905609893}{7} \approx 1.48415158556
So, the solution to the equation ln(7x3)5=3\ln(7x - 3) - 5 = -3 is approximately x1.48415158556x \approx 1.48415158556.

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