Solved on Dec 12, 2023

Solve ln(2x1)=8\ln (2x-1)=8. Round the solution to the nearest thousandth.

STEP 1

Assumptions
1. We are given the equation ln(2x1)=8\ln(2x - 1) = 8.
2. We need to solve for xx.
3. The natural logarithm function is the inverse of the exponential function with base ee.
4. We will round the final answer to the nearest thousandth.

STEP 2

To solve the equation ln(2x1)=8\ln(2x - 1) = 8, we need to rewrite the equation in exponential form to eliminate the natural logarithm.
eln(2x1)=e8e^{\ln(2x - 1)} = e^8

STEP 3

Using the property that eln(a)=ae^{\ln(a)} = a, we can simplify the left side of the equation.
2x1=e82x - 1 = e^8

STEP 4

Now, we need to isolate xx by adding 1 to both sides of the equation.
2x=e8+12x = e^8 + 1

STEP 5

Next, we divide both sides of the equation by 2 to solve for xx.
x=e8+12x = \frac{e^8 + 1}{2}

STEP 6

We calculate the value of e8e^8 using a calculator.
e82980.957987e^8 \approx 2980.957987

STEP 7

Substitute the value of e8e^8 into the equation.
x=2980.957987+12x = \frac{2980.957987 + 1}{2}

STEP 8

Now, perform the addition in the numerator.
x=2980.957987+12=2981.9579872x = \frac{2980.957987 + 1}{2} = \frac{2981.957987}{2}

STEP 9

Finally, divide by 2 to find the value of xx.
x=2981.95798721490.979x = \frac{2981.957987}{2} \approx 1490.979

STEP 10

Round the result to the nearest thousandth.
x1490.979x \approx 1490.979
The solution to the equation ln(2x1)=8\ln(2x - 1) = 8 is approximately x=1490.979x = 1490.979.

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