Questions: ln(4x-1) = 3

Transcript text: $\ln (4 x-1)=3$

Solution

Solution Steps

To solve the equation $\ln(4x - 1) = 3$, we need to eliminate the natural logarithm by exponentiating both sides of the equation. This will allow us to solve for $x$.

Exponentiate both sides to remove the natural logarithm: $4x - 1 = e^3$.
Solve for $x$ by isolating it on one side of the equation.

Step 1: Exponentiate Both Sides

Starting with the equation: \[ \ln(4x - 1) = 3 \] we exponentiate both sides to eliminate the natural logarithm: \[ 4x - 1 = e^3 \]

Step 2: Calculate $e^3$

Calculating $e^3$ gives us: \[ e^3 \approx 20.0855 \] Thus, we can rewrite the equation as: \[ 4x - 1 \approx 20.0855 \]

Step 3: Solve for $x$

Next, we isolate $x$: \[ 4x = 20.0855 + 1 \] \[ 4x \approx 21.0855 \] Dividing both sides by 4: \[ x \approx \frac{21.0855}{4} \approx 5.2714 \]

Final Answer

The solution to the equation is: \[ \boxed{x \approx 5.2714} \]

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