Question

Me ajudem plmds, preciso disso para passar.

Determine o conjunto verdade das equações exponenciais

Answer 1

a) $8^{x}=32$

$(2^3)^{x}=2^{5}$

$2^{3x}=2^{5}$

$3x=5$

$\boxed{x=\dfrac{5}{3}}$

$\text{S}=\{\frac{5}{3}\}$


b) $25^{x}=625$

$(5^2)^{x}=5^{4}$

$5^{2x}=5^{4}$

$2x=4$

$x=\dfrac{4}{2}$

$\boxed{x=2}$

$\text{S}=\{2\}$


c) $9^{x}=\dfrac{1}{3}$

$(3^2)^{x}=3^{-1}$

$3^{2x}=3^{-1}$

$2x=-1$

$\boxed{x=-\dfrac{1}{2}}$

$\text{S}=\{-\frac{1}{2}\}$


d) $\left(\dfrac{9}{25}\right)^{2x}=\dfrac{3}{5}$

$\left[\left(\dfrac{3}{5}\right)^{2}\right]^{2x}=\dfrac{3}{5}$

$\left(\dfrac{3}{5}\right)^{4x}=\left(\dfrac{3}{5}\right)^{1}$

$4x=1$

$\boxed{x=\dfrac{1}{4}}$

$\text{S}=\{\frac{1}{4}\}$


e) $2^{x+4}=16$

$2^{x+4}=2^{4}$

$x+4=4$

$x=4-4$

$\boxed{x=0}$

$\text{S}=\{0\}$


f) $5^{x}=\sqrt{5}$

$5^{x}=5^{\frac{1}{2}}$

$\boxed{x=\dfrac{1}{2}}$

$\text{S}=\{\frac{1}{2}\}$


g) $5^{2x+1}=\dfrac{1}{625}$

$5^{2x+1}=5^{-4}$

$2x+1=-4$

$2x=-5$

$\boxed{x=-\dfrac{5}{2}}$

$\text{S}=\{-\frac{5}{2}\}$


h) $25^{x+2}=1$

$25^{x+2}=25^{0}$

$x+2=0$

$\boxed{x=-2}$

$\text{S}=\{-2\}$


i) $0,01^{9x-1}=0,01^{0}$

$9x-1=0$

$9x=1$

$\boxed{x=\dfrac{1}{9}}$

$\text{S}=\{\frac{1}{9}\}$


j) $32^{x+3}=\sqrt[3]{2}$

$(2^5)^{x+3}=2^{\frac{1}{3}}$

$2^{5x+15}=2^{\frac{1}{3}}$

$5x+15=\dfrac{1}{3}$

$15x+45=1$

$15x=-44$

$\boxed{x=-\dfrac{44}{15}}$

$\text{S}=\{-\frac{44}{15}\}$


k) $5^{x^2+2x}=1$

$5^{x^2+2x}=5^{0}$

$x^2+2x=0$

$x\cdot(x+2)=0$

$\boxed{x'=0}$

$x+2=0 \iff \boxed{x"=-2}$

$\text{S}=\{-2,0\}$