Resposta:

[tex]\Large \textsf{Leia abaixo}[/tex]

Explicação passo a passo:

[tex]\Large \text{$ \sf 5^{x - 2} - \dfrac{5^{-x}}{5^{-3}} = 100$}[/tex]

[tex]\Large \text{$ \sf \dfrac{5^x}{5^2} - \dfrac{5^{3}}{5^{x}} = 100$}[/tex]

[tex]\Large \text{$ \sf 5^{2x} - 5^{5} = 100\:.\:5^2\:.\:5^x$}[/tex]

[tex]\Large \text{$ \sf 5^{2x} - 2.500\:.\:5^x - 3.125 = 0$}[/tex]

[tex]\Large \text{$ \sf y = 5^{x}$}[/tex]

[tex]\Large \text{$ \sf y^{2} - 2.500y - 3.125 = 0$}[/tex]

[tex]\Large \text{$ \sf a = 1 \leftrightarrow b = -2.500 \leftrightarrow c = -3.125 $}[/tex]

[tex]\Large \text{$ \sf \Delta = b^2 - 4.a.c $}[/tex]

[tex]\Large \text{$ \sf \Delta = (-2.500)^2 - 4.1.(-3.125) $}[/tex]

[tex]\Large \text{$ \sf \Delta = 6.250.000 + 12.500 $}[/tex]

[tex]\Large \text{$ \sf \Delta = 6.262.500 $}[/tex]

[tex]\Large \text{$ \sf \sf{x = \dfrac{-b \pm \sqrt{\Delta}}{2a} = \dfrac{2.500 \pm \sqrt{6.262.500}}{2} \rightarrow \begin{cases}\sf{x' = 2.501,25}\\\\\sf{x'' = -2,50}\end{cases}}$}[/tex]

[tex]\Large \text{$ \sf 5^x = 2.501,25$}[/tex]

[tex]\Large \text{$ \sf log_5\:5^x = log_5\:2.501,25$}[/tex]

[tex]\Large \text{$ \sf x = log_5\:2.501,25$}[/tex]

[tex]\Large \boxed{\boxed{\text{$ \sf x \approx 4,86$}}}[/tex]