Questions: Solve for x. ln 2 + ln (x+1) = -3

Solve for x. ln 2 + ln (x+1) = -3
Transcript text: Solve for $x$. \[ \ln 2+\ln (x+1)=-3 \]
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Solution

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Solution Steps

To solve the equation \(\ln 2 + \ln (x+1) = -3\), we can use the properties of logarithms to combine the logarithmic terms and then exponentiate both sides to solve for \(x\).

  1. Use the property of logarithms: \(\ln a + \ln b = \ln (a \cdot b)\).
  2. Combine the logarithmic terms: \(\ln (2(x+1)) = -3\).
  3. Exponentiate both sides to remove the natural logarithm: \(2(x+1) = e^{-3}\).
  4. Solve for \(x\).
Step 1: Combine Logarithmic Terms

Given the equation: \[ \ln 2 + \ln (x+1) = -3 \] we use the property of logarithms \(\ln a + \ln b = \ln (a \cdot b)\) to combine the terms: \[ \ln (2(x+1)) = -3 \]

Step 2: Exponentiate Both Sides

Exponentiate both sides to remove the natural logarithm: \[ 2(x+1) = e^{-3} \] Given \(e^{-3} \approx 0.04979\), we have: \[ 2(x+1) = 0.04979 \]

Step 3: Solve for \(x\)

Divide both sides by 2: \[ x+1 = \frac{0.04979}{2} \] \[ x+1 \approx 0.02489 \] Subtract 1 from both sides: \[ x \approx 0.02489 - 1 \] \[ x \approx -0.9751 \]

Step 4: Round to the Nearest Hundredth

Round the answer to the nearest hundredth: \[ x \approx -0.98 \]

Final Answer

\[ \boxed{x = -0.98} \]

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