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L&E, 2ADV E1 EQ-Bank 3

Solve the equation  \(4^x-2^{x+2}=32\), showing all working.   (3 marks)

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\(x=3\)

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  \(2^{2 x}-2^{x+2}\) \(=2^5\)
  \(2^{2 x}-2^2 \times 2^x-32\) \(=0\)

 
\(\text{Let} \ \ 2^x=X\)

  \(X^2-4X-32\) \(=0\)
  \((X-8)(X+4)\) \(=0\)

\(X\) \(=8\) \(\quad\text{or}\quad\) \(X\) \(=-4\)
\(2^x\) \(=8\)   \(2^x\) \(=-4\ \ \text{(no solution)}\)
\(x\) \(=3\)      

 
\(\therefore x=3\)

L&E, 2ADV E1 2023 MET1 2

Solve  \(e^{2x}-12=4e^{x}\)  for  \(x\ \in\ R\).   (3 marks)

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\(x=\log_{e}6\)

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\(e^{2x}-12\) \(=4e^{x}\)
\(e^{2x}-4e^{x}-12\) \(=0\)

 
\(\text{Let}\ \ u=e^{x}:\)

\(u^2-4u-12\) \(=0\)
\((u-6)(u+2)\) \(=0\)
 

\(\Rightarrow u=6\ \ \ \text{or}\ -2\)

\(\therefore e^{x}\) \(=6\ \ \ \ \ \ \ \ \ \ \text{or}\ \ \ \ \ \ e^{x}=-2\ \text{(no solution)}\)  
\(x\) \(=\log_{e}6 \)  

L&E, 2ADV E1 2019 HSC 15a

Solve  `e^(2 ln x) = x + 6`   (2 marks)

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`x = 3 or -2`

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♦ Mean mark 47%.

`e^(2 ln x)` `= x + 6`
`ln e^(2 ln x)` `= ln (x + 6)`
`2 ln x` `= ln (x + 6)`
`ln x^2` `= ln (x + 6)`
`x^2` `= x + 6`
`x^2-x-6` `= 0`
`(x-3) (x + 2)` `= 0`

 
`:. x = 3 \ \ (x>0)`

L&E, 2ADV E1 EQ-Bank 9 MC

Solve the equation  `e^(4x)-5e^(2x) + 4 = 0`  for `x`

  1. `x= 1, 4`
  2. `x= – 4, – 1`
  3. `x= 0, log_e 2`
  4. `x= – log_e 2, 0, log_e 2 `
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`C`

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`e^(4x)-5e^(2x) + 4 = 0`

`text(Let)\ \ X=e^(2x)`

`X^2-5X+4` `=0`
`(X-4)(X-1)` `=0`
`X` `=4 or 1`

 

`:.e^(2x)` `=4` `e^(2x)` `=1`
`2x` `=log_e 4` `x` `=0`
`x` `=(2log_e 2)/2`    
  `=log_e 2`    

`=> C`

L&E, 2ADV E1 2004 HSC 6a

Solve the following equation for `x`:

`e^(2x) + 3e^x-10 = 0`.  (2 marks)

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`x = ln 2`

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`e^(2x) + 3e^x-10` `= 0`
`:. (e^x)^2 + 3e^x-10` `= 0`

 
`text(Let)\ \ X = e^x,`

`:. X^2 + 3X-10` `= 0`
`(X + 5)(X-2)` `= 0`
`:. X =2 or -5`

 

`text(If)\ \ X` `=2`
`e^x` `=2`
`ln e^x` `=ln 2`
`x` `=ln 2`
`text(If)\ \ X` `=-5`
`e^x` `=-5\ \ \ text{(no solution)}`

 
 `:. x=ln 2`

L&E, 2ADV E1 2007 HSC 6a

Solve the following equation for `x`:

`2e^(2x)-e^x = 0`.   (2 marks)

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`x = ln\ 1/2`

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`text(Solution 1)`

`2e^(2x)-e^x = 0`

`text(Let)\ \ X = e^x:`

`2X^2-X` `= 0`
`X (2X-1)` `= 0`

 
`X = 0 or 1/2`

 
`text(When)\ e^x = 0\  =>\ text(no solution)`

`text(When)\ e^x = 1/2`

`ln e^x` `= ln\ 1/2`
`:. x` `= ln\ 1/2`

 

`text(Solution 2)`

`2e^(2x)-e^x` `=0`
`2e^(2x)` `=e^x`
`ln 2e^(2x)` `=ln e^x`
`ln 2 +ln e^(2x)` `=x`
`ln 2 + 2x` `=x`
`x` `=-ln2`
  `=ln\ 1/2`