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Opgave 1

$
\eqalign{
& {}^3\log \left( {2x^2 - 3} \right) = 6 \cr
& 2x^2 - 3 = 3^6 \cr
& 2x^2 - 3 = 729 \cr
& 2x^2 = 732 \cr
& x^2 = 366 \cr
& x = - \sqrt {366} \,\,of\,\,x = \sqrt {366} \cr}
$

$
\eqalign{
& {}^{\frac{1}
{2}}\log \left( {\frac{1}
{{4x}}} \right) = 4 \cr
& \frac{1}
{{4x}} = \left( {\frac{1}
{2}} \right)^4 \cr
& \frac{1}
{{4x}} = \frac{1}
{{16}} \cr
& 4x = 16 \cr
& x = 4 \cr}
$

$
\eqalign{
& {}^2\log \left( {4 - 30x^2 } \right) = - 2 \cr
& 4 - 30x^2 = 2^{ - 2} \cr
& 4 - 30x^2 = \frac{1}
{4} \cr
& 16 - 120x^2 = 1 \cr
& 120x^2 = 15 \cr
& x^2 = \frac{1}
{8} \cr
& x = - \sqrt {\frac{1}
{8}} \,\,of\,\,x = \sqrt {\frac{1}
{8}} \cr
& x = - \frac{1}
{4}\sqrt 2 \,\,of\,\,x = \frac{1}
{4}\sqrt 2 \cr}
$

Opgave 2

$3^{x+1}=80$
$x+1=^3log(80)$
$x=-1+^3log(80)$

$5+2^{3x}=25$
$2^{3x}=20$
$3x=^2log(20)$
$x=\frac{1}{3}·^2log(20)$

$\eqalign{4-log(\frac{1}{x})=2}$
$\eqalign{-log(\frac{1}{x})=-2}$
$\eqalign{log(\frac{1}{x})=2}$
$\eqalign{\frac{1}{x}=100}$
$\eqalign{x=\frac{1}{100}}$

Opgave 3

$
\begin{array}{l}
y = 25 - 5^{\frac{1}{2}x + 2} \\
y - 25 = - 5^{\frac{1}{2}x + 2} \\
- y + 25 = 5^{\frac{1}{2}x + 2} \\
\frac{1}{2}x + 2 = {}^5\log \left( { - y + 25} \right) \\
\frac{1}{2}x = {}^5\log \left( { - y + 25} \right) - 2 \\
x = 2 \cdot {}^5\log \left( { - y + 25} \right) - 4 \\
\end{array}
$

$
\begin{array}{l}
y = 3 \cdot 2^x + 5 \\
3 \cdot 2^x = y - 5 \\
2^x = \frac{1}{3}y - 1\frac{2}{3} \\
x = {}^2\log \left( {\frac{1}{3}y - 1\frac{2}{3}} \right) \\
\end{array}
$

of...

$
\begin{array}{l}
y = 3 \cdot 2^x + 5 \\
3 \cdot 2^x = y - 5 \\
2^x = \frac{{y - 5}}{3} \\
x = {}^2\log \left( {\frac{{y - 5}}{3}} \right) \\
\end{array}
$

$ \begin{array}{l} y = 10^{x^2 } - 1 \\ 10^{x^2 } = y + 1 \\ x^2 = \log (y + 1) \\ x = \sqrt {\log (y + 1)} \\ \end{array} $

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