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Find the exact solutions to the equations (a) ln x + ln 3 = ln 6, (b) e^x + 3e^x = 4. - Edexcel - A-Level Maths Pure - Question 3 - 2007 - Paper 5
Question 3
Find the exact solutions to the equations (a) ln x + ln 3 = ln 6, (b) e^x + 3e^x = 4.
Worked Solution & Example Answer:Find the exact solutions to the equations (a) ln x + ln 3 = ln 6, (b) e^x + 3e^x = 4. - Edexcel - A-Level Maths Pure - Question 3 - 2007 - Paper 5
Step 1
(a) ln x + ln 3 = ln 6
Answer
To solve the equation, we can use the properties of logarithms.
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Use the property that states: ( \ln a + \ln b = \ln( a \cdot b ) )
So we can rewrite the left-hand side:
[ \ln(x) + \ln(3) = \ln(6) \implies \ln(3x) = \ln(6) ]
- By exponentiating both sides, we eliminate the logarithm:
[ 3x = 6 ]
- Now solve for (x):
[ x = \frac{6}{3} = 2 ]
Thus, the exact solution is (x = 2).
Step 2
(b) e^x + 3e^x = 4
Answer
We start by simplifying the left side:
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Combine like terms:
[ e^x + 3e^x = 4 \implies 4e^x = 4 ]
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Divide both sides by 4:
[ e^x = 1 ]
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To eliminate the exponential, take the natural logarithm of both sides:
[ \ln(e^x) = \ln(1) \implies x = 0 ]
Thus, the exact solution is (x = 0).