| 1. | $
\eqalign{
& e^{2x} - 4e^x = 45 \cr
& (e^x )^2 - 4e^x - 45 = 0 \cr
& y = e^x \cr
& y^2 - 4y - 45 = 0 \cr
& (y - 9)(y + 5) = 0 \cr
& y = 9 \vee y = - 5 \cr
& e^x = 9 \vee e^x = - 5\,\,(v.n.) \cr
& x = 2\ln (3) \cr}
$
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| 2. | $
\eqalign{
& 3^x \cdot 3^{x + 2} = 9 \cr
& 3^{x + x + 2} = 3^2 \cr
& 3^{2x + 2} = 3^2 \cr
& 2x + 2 = 2 \cr
& 2x = 0 \cr
& x = 0 \cr}
$
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| 3. | $
\eqalign{
& x \cdot 2^{2x} = x \cdot \sqrt 8 \cr
& x = 0 \vee 2^{2x} = 2^{1\frac{1}
{2}} \cr
& x = 0 \vee 2x = 1\frac{1}
{2} \cr
& x = 0 \vee x = \frac{3}
{4} \cr}
$
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| 4. | $
\eqalign{
& x \cdot e^{x + 1} - e^x = 0 \cr
& x \cdot e \cdot e^x - e^x = 0 \cr
& e^x \left( {x \cdot e - 1} \right) = 0 \cr
& e^x = 0\,\,(v.n.) \vee x \cdot e - 1 = 0 \cr
& x \cdot e = 1 \cr
& x = \frac{1}
{e} \cr}
$
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| 5. | $
\eqalign{
& 4x\left( {2^x - 3} \right) = 0 \cr
& 4x = 0 \vee 2^x - 3 = 0 \cr
& x = 0 \vee 2^x = 3 \cr
& x = 0 \vee x = {}^2\log (3) \cr}
$
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