Question

Resolva, em R, as seguintes equações exponenciais..

a) 3x= 81
b) 2x=256
c) 7x=7
d) 5x+2=125
e) 8x=16
f) 27x=9
g) 4x=32
h) 25x=625

Answer 1

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a)

$\sf3^x=81\\\begin{array}{c|l}\sf81&\sf3\\\sf27&\sf3\\\sf9&3\\\sf3&\sf3\\\sf1\end{array}\\\sf 81=3^4\\\sf 3^x=3^4\\\huge\boxed{\boxed{\boxed{\boxed{\sf x=4}}}}$

b)

$\sf2^x=256\\\begin{array}{c|l}\sf256&\sf2\\\sf128&\sf2\\\sf64&\sf2\\\sf32&\sf2\\\sf16&\sf2\\\sf8&\sf2\\\sf4&\sf2\\\sf2&\sf2\\\sf1\end{array}\\\sf256=2^8\\\sf 2^x=2^8\\\huge\boxed{\boxed{\boxed{\boxed{\sf x=8}}}}$

c)

$\sf7^x=7\\\sf 7^x=7^1\\\huge\boxed{\boxed{\boxed{\boxed{\sf x=1}}}}$

d)

$\sf5^{x+2}=125\\\begin{array}{c|l}\sf125&\sf5\\\sf25&\sf5\\\sf5&\sf5\\\sf1\end{array}\\\sf 125=5^3\\\sf 5^{x+2}=5^3\\\sf x+2=3\\\sf x=3-2\\\huge\boxed{\boxed{\boxed{\boxed{\sf x=1}}}}$

e)

$\sf8^x=16\\\begin{array}{c|l}\sf16&\sf2\\\sf8&\sf2\\\sf4&\sf2\\\sf2&\sf2\\\sf1\end{array}\\\sf 16=2^4~~~8=2^3\\\sf (2^3)^x=2^4\\\sf 2^{3x}=2^4\\\sf 3x=4\\\huge\boxed{\boxed{\boxed{\boxed{\sf x=\dfrac{4}{3}}}}}$

f)

$\sf27^x=9\\\begin{array}{c|l}\sf27&\sf3\\\sf9&\sf3\\\sf3&\sf3\\\sf1\end{array}\\\sf 27=3^3~~~9=3^2\\\sf (3^3)^x=3^2\\\sf 3^{3x}=3^2\\\sf 3x=2\\\huge\boxed{\boxed{\boxed{\boxed{\sf x=\dfrac{2}{3}}}}}$

g)

$\sf4^x=32\\\begin{array}{c|l}\sf32&\sf2\\\sf16&\sf2\\\sf8&\sf2\\\sf4&\sf2\\\sf2&\sf2\\\sf1\end{array}\\\sf 32=2^5~~~4=2^2\\\sf (2^2)^x=2^5\\\sf2^{2x}=2^5\\\sf 2x=5\\\huge\boxed{\boxed{\boxed{\boxed{\sf x=\dfrac{5}{2}}}}}$

h)

$\sf25^x=625\\\begin{array}{c|l}\sf625&\sf5\\\sf125&\sf5\\\sf25&\sf5\\\sf5&\sf5\\\sf1\end{array}\\\sf 625=5^4~~~25=5^2\\\sf (5^2)^x=5^4\\\sf 5^{2x}=5^2\\\sf 2x=4\\\sf x=\dfrac{4}{2}\\\huge\boxed{\boxed{\boxed{\boxed{\sf x=2}}}}$