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Sagot :
To solve the equation [tex]\(2 \ln (x-3) = -4\)[/tex], we need to follow these steps:
1. Isolate the natural logarithm (ln) term:
[tex]\[ 2 \ln (x-3) = -4 \][/tex]
Divide both sides by 2:
[tex]\[ \ln (x-3) = -2 \][/tex]
2. Exponentiate both sides to eliminate the natural logarithm:
Recall that [tex]\(e^{\ln a} = a\)[/tex]. Thus, applying the exponential function to both sides:
[tex]\[ e^{\ln (x-3)} = e^{-2} \][/tex]
[tex]\[ x-3 = e^{-2} \][/tex]
3. Solve for [tex]\(x\)[/tex]:
Add 3 to both sides to isolate [tex]\(x\)[/tex]:
[tex]\[ x = e^{-2} + 3 \][/tex]
Now to compute the numerical value:
[tex]\[ e^{-2} \approx 0.1353352832366127 \][/tex]
Then,
[tex]\[ x = 0.1353352832366127 + 3 \][/tex]
[tex]\[ x \approx 3.135335283236613 \][/tex]
4. Round the final answer to the nearest hundredth:
[tex]\[ x \approx 3.14 \][/tex]
Therefore, the solution to the equation [tex]\(2 \ln (x-3) = -4\)[/tex] rounded to the nearest hundredth is:
[tex]\[ x \approx 3.14 \][/tex]
1. Isolate the natural logarithm (ln) term:
[tex]\[ 2 \ln (x-3) = -4 \][/tex]
Divide both sides by 2:
[tex]\[ \ln (x-3) = -2 \][/tex]
2. Exponentiate both sides to eliminate the natural logarithm:
Recall that [tex]\(e^{\ln a} = a\)[/tex]. Thus, applying the exponential function to both sides:
[tex]\[ e^{\ln (x-3)} = e^{-2} \][/tex]
[tex]\[ x-3 = e^{-2} \][/tex]
3. Solve for [tex]\(x\)[/tex]:
Add 3 to both sides to isolate [tex]\(x\)[/tex]:
[tex]\[ x = e^{-2} + 3 \][/tex]
Now to compute the numerical value:
[tex]\[ e^{-2} \approx 0.1353352832366127 \][/tex]
Then,
[tex]\[ x = 0.1353352832366127 + 3 \][/tex]
[tex]\[ x \approx 3.135335283236613 \][/tex]
4. Round the final answer to the nearest hundredth:
[tex]\[ x \approx 3.14 \][/tex]
Therefore, the solution to the equation [tex]\(2 \ln (x-3) = -4\)[/tex] rounded to the nearest hundredth is:
[tex]\[ x \approx 3.14 \][/tex]
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