rozwiąż równanie a)[latex] log_{x}27=3 [/latex] b) [latex] log_{x-4}2x=2[/latex]

[latex]a) log_{x}27=3 Zal.\ 27=x^3 x extgreater 0 i x ot=1\ x^3=3^3\ oxed{x=3}[/latex] [latex]b) log_{x-4}2x=2 Zal.\ 2x=(x-4)^2 x-4 extgreater 0 i x-4 ot=1 i x extgreater 0\ 2x=x^2-8x+16 x extgreater 4 i x ot=5 i x extgreater 0\ x^2-10x+16=0\ Delta=100-64=36\ sqrt{Delta}=6\ \ x_1=frac{10-6}{2}=2\ x_2=frac{10+6}{2}=8\ \ \ oxed{x=8}[/latex]

[latex]log_{a}b = c, extless = extgreater a^{c} = b[/latex] [latex]a,b extgreater 0 oraz a eq 1[/latex] [latex]a)\log_{x}27 = 3[/latex] Z: x > 0 [latex]x^{3}=27\\x^{3}=3^{3}\\x = 3[/latex] [latex]b)\log_{x-4}2x = 2[/latex] Z: 2x > 0 x > 0 x - 4 > 0 x > 4 i x - 4 ≠ 1 x ≠ 5 [latex](x-4)^{2}=2x\\x^{2}-8x+16=2x\\x^{2}-10x+16 = 0\\Delta = 100-64 = 36\\sqrt{Delta} = 6[/latex] [latex]x_1 = frac{10-6}{2} = 2 ot in D, x_2 = frac{10+6}{2}=8\\x = 8[/latex]