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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

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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

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Answer

To solve the equation, we use the property of logarithms that states: ( \ln a + \ln b = \ln(ab)). Thus, we can rewrite the left side as:

\ln(x \cdot 3) = \ln(6)

This leads to:

3x = 6.

Dividing both sides by 3 gives:

x = 2.

This is the exact solution for part (a).

Step 2

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Answer

First, we can combine the terms on the left:

4e^x = 4.

Dividing both sides by 4 results in:

e^x = 1.

Taking the natural logarithm of both sides:

x = \ln(1) = 0.

Thus, the exact solution for part (b) is:

x = 0.

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