Solved on Dec 12, 2023

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

STEP 1

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

STEP 2

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

STEP 3

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

STEP 4

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

STEP 5

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

STEP 6

We calculate the value of $e^8$ using a calculator.
$e^8 \approx 2980.957987$

STEP 7

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

STEP 8

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

STEP 9

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

STEP 10

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

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

STEP 1

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

STEP 2

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

STEP 3

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

STEP 4

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

STEP 5

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

STEP 6

We calculate the value of $e^8$ using a calculator.
$e^8 \approx 2980.957987$

STEP 7

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

STEP 8

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

STEP 9

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

STEP 10

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

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

STEP 1

Assumptions1. We are given the equation ln⁡(2x−1)=8\ln(2x - 1) = 8ln(2x−1)=8.2. We need to solve for xxx.3. The natural logarithm function is the inverse of the exponential function with base eee.4. We will round the final answer to the nearest thousandth.

STEP 2

To solve the equation ln⁡(2x−1)=8\ln(2x - 1) = 8ln(2x−1)=8, we need to rewrite the equation in exponential form to eliminate the natural logarithm.eln⁡(2x−1)=e8e^{\ln(2x - 1)} = e^8eln(2x−1)=e8

STEP 3

Using the property that eln⁡(a)=ae^{\ln(a)} = aeln(a)=a, we can simplify the left side of the equation.2x−1=e82x - 1 = e^82x−1=e8

STEP 4

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

STEP 5

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

STEP 6

We calculate the value of e8e^8e8 using a calculator.e8≈2980.957987e^8 \approx 2980.957987e8≈2980.957987

STEP 7

Substitute the value of e8e^8e8 into the equation.x=2980.957987+12x = \frac{2980.957987 + 1}{2}x=22980.957987+1​

STEP 8

Now, perform the addition in the numerator.x=2980.957987+12=2981.9579872x = \frac{2980.957987 + 1}{2} = \frac{2981.957987}{2}x=22980.957987+1​=22981.957987​

STEP 9

Finally, divide by 2 to find the value of xxx.x=2981.9579872≈1490.979x = \frac{2981.957987}{2} \approx 1490.979x=22981.957987​≈1490.979

STEP 10

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

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

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

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

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

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

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

We calculate the value of $e^8$ using a calculator.
$e^8 \approx 2980.957987$

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

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

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

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