Answer :
To solve the equation [tex]\( 19 + 2 \ln(x) = 25 \)[/tex], let's follow the steps:
1. Isolate the logarithmic term:
[tex]\[ 19 + 2 \ln(x) = 25 \][/tex]
Subtract 19 from both sides:
[tex]\[ 2 \ln(x) = 25 - 19 \][/tex]
Simplify the right-hand side:
[tex]\[ 2 \ln(x) = 6 \][/tex]
2. Solve for [tex]\( \ln(x) \)[/tex]:
Divide both sides by 2:
[tex]\[ \ln(x) = \frac{6}{2} \][/tex]
Simplify the fraction:
[tex]\[ \ln(x) = 3 \][/tex]
3. Exponentiate both sides to solve for [tex]\( x \)[/tex]:
To eliminate the natural logarithm, exponentiate both sides with base [tex]\( e \)[/tex]:
[tex]\[ e^{\ln(x)} = e^3 \][/tex]
Since [tex]\( e^{\ln(x)} = x \)[/tex], we have:
[tex]\[ x = e^3 \][/tex]
4. Calculate the value of [tex]\( e^3 \)[/tex]:
Using a known value for [tex]\( e \)[/tex], where [tex]\( e \approx 2.718 \)[/tex]:
[tex]\[ e^3 \approx 2.718^3 \approx 20.086 \][/tex]
5. Select the closest answer:
The given choices are:
- A. [tex]\( x \approx 20.09 \)[/tex]
- B. [tex]\( x \approx 1.93 \)[/tex]
- C. [tex]\( x \approx 3 \)[/tex]
- D. [tex]\( x \approx 0.05 \)[/tex]
The calculated value [tex]\( \approx 20.086 \)[/tex] is closest to option A: [tex]\( x \approx 20.09 \)[/tex].
Hence, the correct answer is:
[tex]\[ \boxed{x \approx 20.09} \][/tex]
1. Isolate the logarithmic term:
[tex]\[ 19 + 2 \ln(x) = 25 \][/tex]
Subtract 19 from both sides:
[tex]\[ 2 \ln(x) = 25 - 19 \][/tex]
Simplify the right-hand side:
[tex]\[ 2 \ln(x) = 6 \][/tex]
2. Solve for [tex]\( \ln(x) \)[/tex]:
Divide both sides by 2:
[tex]\[ \ln(x) = \frac{6}{2} \][/tex]
Simplify the fraction:
[tex]\[ \ln(x) = 3 \][/tex]
3. Exponentiate both sides to solve for [tex]\( x \)[/tex]:
To eliminate the natural logarithm, exponentiate both sides with base [tex]\( e \)[/tex]:
[tex]\[ e^{\ln(x)} = e^3 \][/tex]
Since [tex]\( e^{\ln(x)} = x \)[/tex], we have:
[tex]\[ x = e^3 \][/tex]
4. Calculate the value of [tex]\( e^3 \)[/tex]:
Using a known value for [tex]\( e \)[/tex], where [tex]\( e \approx 2.718 \)[/tex]:
[tex]\[ e^3 \approx 2.718^3 \approx 20.086 \][/tex]
5. Select the closest answer:
The given choices are:
- A. [tex]\( x \approx 20.09 \)[/tex]
- B. [tex]\( x \approx 1.93 \)[/tex]
- C. [tex]\( x \approx 3 \)[/tex]
- D. [tex]\( x \approx 0.05 \)[/tex]
The calculated value [tex]\( \approx 20.086 \)[/tex] is closest to option A: [tex]\( x \approx 20.09 \)[/tex].
Hence, the correct answer is:
[tex]\[ \boxed{x \approx 20.09} \][/tex]
