Question

PF ME AJUDA URGENTE. RELACIONE AS SEGUINTES EQUAÇÕES EXPONENCIAIS NA VARIÁVEL X COM SUAS RESPECTIVAS SOLUÇÕES:

EQUAÇÕES EXPONENCIAIS/SOLUÇÃO
27ˣ=243 ( )
7²ˣ⁻⁷=343 ( )
100⁵⁺ˣ=1/10000 ( )
4ˣ²⁻²ˣ⁻⁵=64 ( )
6³ˣ⁻ˣ²=36 ( )
-1/3ˣ⁴=7 ( )
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(1) S= {-7}
(2) S= {2,4}
(3) S= {5/3}
(4) S= {1,2}
(5) S= {-2}
(6) S= {-4}
(7) S= {5}
(8) S= {-5}

Answer 1

• Resolvendo $27^x = 243$:

$log_3(27^x) = log_3(243) \\x \cdot log_3(27) = log_3(243) \\x \cdot log_3(3^3) = log_3(3^5) \\3x \cdot log_3(3) = 5 \cdot log_3(3) \\3x = 5 \\\\x = \dfrac{5}{3} = \text{n\'umero (3)}$

• Resolvendo $7^{2x-7} = 343$:

$log_7(7^{2x-7}) = log_7(343) \\(2x - 7) \cdot log_7(7) = log_7(7^3) \\(2x - 7) \cdot log_7(7) = 3 \cdot log_7(7) \\2x - 7 = 3\\\\x = \dfrac{7+3}{2} = 5 = \text{n\'umero (7)}$

• Resolvendo $100^{5+x}= \dfrac{1}{10000}$:

$100^{5+x}=10000^{-1} \\log_{100}(100^{5+x}) = log_{100}(10000^{-1}) \\(5+x) \cdot log_{100}(100) = -1 \cdot log_{100}(10000) \\(5+x) \cdot log_{100}(100) = -1 \cdot log_{100}(100^2) \\(5+x) \cdot log_{100}(100) = -1 \cdot 2 \cdot log_{100}(100) \\5 + x = -2 \\x = -2 -5 = -7 = \text{n\'umero (1)}$

• Resolvendo $4^{x^2-2x-5}=64$:

$log_4(4^{x^2-2x-5})=log_4(64) \\(x^2-2x-5) \cdot log_4(4) = log_4(64) \\(x^2-2x-5) \cdot log_4(4) = log_4(4^3) \\(x^2-2x-5) \cdot log_4(4) = 3 \cdot log_4(4) \\x^2 - 2x - 5 = 3 \\x^2 - 2x - 8 = 0 \\\\\Delta = b^2 - 4 \cdot a \cdot c \\\Delta = 4 + 4 \cdot 1 \cdot 8 \\\Delta = 4 + 32 = 36 \\\\x_1 = \dfrac{-b+\sqrt{\Delta}}{2\cdot a} = \dfrac{2+6}{2} = 4 \\\\x_2 = \dfrac{-b-\sqrt{\Delta}}{2\cdot a} = \dfrac{2-6}{2} = -2 \\\\ \text{n\'umero (2)}$

• Resolvendo $6^{3x-x^2} =36$:

$log_6(6^{3x-x^2}) = log_6(36) \\(3x - x^2) \cdot log_6(6) = log_6(36) \\(3x - x^2) \cdot log_6(6) = log_6(6^2) \\(3x - x^2) \cdot log_6(6) = 2 \cdot log_6(6) \\3x - x^2 = 2\\-x^2 + 3x - 2 = 0 \\\\\Delta = b^2 - 4 \cdot a \cdot c \\\Delta = 9 - 4 \cdot 1 \cdot 2 \\\Delta = 9 - 8 = 1 \\\\x_1 = \dfrac{-b+\sqrt{\Delta}}{2\cdot a} = \dfrac{-3+1}{-2} = 1 \\\\x_2 = \dfrac{-b-\sqrt{\Delta}}{2\cdot a} = \dfrac{-3-1}{-2} = 2 \\\\ \text{n\'umero (4)}$

• Resolvendo $\left( -\dfrac{1}{3} \right) ^{x^4} = 7$:

Não há soluções no conjunto dos reais para esse problema.