4.5\nscore: 20/24 answered: 20/24\nquestion 21\nsolve the equation for ( x ). answer exactly.\n( 4 ln (x +…

Explanation:

Step1: Isolate the natural logarithm term

Divide both sides of the equation (4\ln(x + 1)=12) by (4). [ \begin{align*} \frac{4\ln(x + 1)}{4}&=\frac{12}{4}\ \ln(x + 1)& = 3 \end{align*} ]

Step2: Use the property of logarithms

Since (\ln a=b) is equivalent to (a = e^{b}) (where (a=x + 1) and (b = 3)), we can rewrite the equation. [x+1=e^{3}]

Step3: Solve for (x)

Subtract (1) from both sides of the equation (x+1=e^{3}). [x=e^{3}-1]

Answer:

(x = e^{3}-1)