Odpowiedź:
12]
log √3 2-log √3 x =4 log √3 [ 2/x]=4 √3⁴=2/x
2/x=9 9x=2 x= 2/9
13]
log ₂20= log ₂[2*10]= log ₂ 2 + log ₂ 10= 1 + log ₂10= 1 + log 10/ log 2= 1 + 1/0,3=1 + 1: 3/10=1 + 10/3= 13/3
log ₂₀2= log 2/log 20= log 2 / log (2*10]= log 2/ [ log 2 + log 10]=
0,3 / [ 0,3+ 1]= 0,3/1,3= 3/13
całosc = 13/3-3/13=169/39-9/39=160/39
14]
x>0
2 log [5x]²+1/2 log x - 3 log [x/2]= log [ 25x²]²+log x¹/²- log [1/2 x ]³=
log 625x⁴+log x¹/²- log [1/8 x ³]= log [ 625 x ⁴* x ¹/² : x³/8]=
log [625x ⁹/² *8/x³]= log 5000 x ⁹/²⁻³= log 5000 x ³/²
15]
log ₂ n/3- log ₂ 3+ log ₂n²= log ₂[ n/3 :3 *n²]= log ₂[ n³/9]
Szczegółowe wyjaśnienie:
12.
[tex]log_{\sqrt{3}}2-log_{\sqrt{3}}x=4\\\\log_{\sqrt{3}}\frac{2}{x}=4\\\\\sqrt{3}^4=\frac{2}{x}\\\\9=\frac{2}{x}\rightarrow x=\frac{2}{9}\\[/tex]
13.
[tex]log_2{20}-log_{20}2=\\\\(log_2{2}+log_2{10})-(log_{20}20-log_{20}10)=\\\\(1+log_2{10})-(1-log_{20}10)=\\\\(1+\frac{log10}{log2})-(1-\frac{log10}{log20})=\\\\(1+\frac{1}{0,3})-(1-\frac{1}{log10+log2})=\\\\4\frac{1}{3}-(1-\frac{1}{1+0,3})=\\\\4\frac{1}{3}-\frac{3}{13}=\\\\\frac{160}{39}\\[/tex]
14.
[tex]2log(5x)^2+\frac{1}{2}logx-3log\frac{x}{2}=\\\\log(5x)^4+logx^{\frac{1}{2}}-log(\frac{x}{2})^3=\\\\log\frac{(5x)^4(x)^{\frac{1}{2} }}{(\frac{x}{2})^3}\\[/tex]
15.
[tex]log_{2}\frac{n}{3}-log_{2}3+log_{2}n^2\\\\log_{2}\frac{\frac{n}{3}}{3}*n^2\\\\log_{2}\frac{n^3}{9}\\[/tex]
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Odpowiedź:
12]
log √3 2-log √3 x =4 log √3 [ 2/x]=4 √3⁴=2/x
2/x=9 9x=2 x= 2/9
13]
log ₂20= log ₂[2*10]= log ₂ 2 + log ₂ 10= 1 + log ₂10= 1 + log 10/ log 2= 1 + 1/0,3=1 + 1: 3/10=1 + 10/3= 13/3
log ₂₀2= log 2/log 20= log 2 / log (2*10]= log 2/ [ log 2 + log 10]=
0,3 / [ 0,3+ 1]= 0,3/1,3= 3/13
całosc = 13/3-3/13=169/39-9/39=160/39
14]
x>0
2 log [5x]²+1/2 log x - 3 log [x/2]= log [ 25x²]²+log x¹/²- log [1/2 x ]³=
log 625x⁴+log x¹/²- log [1/8 x ³]= log [ 625 x ⁴* x ¹/² : x³/8]=
log [625x ⁹/² *8/x³]= log 5000 x ⁹/²⁻³= log 5000 x ³/²
15]
log ₂ n/3- log ₂ 3+ log ₂n²= log ₂[ n/3 :3 *n²]= log ₂[ n³/9]
Szczegółowe wyjaśnienie:
12.
[tex]log_{\sqrt{3}}2-log_{\sqrt{3}}x=4\\\\log_{\sqrt{3}}\frac{2}{x}=4\\\\\sqrt{3}^4=\frac{2}{x}\\\\9=\frac{2}{x}\rightarrow x=\frac{2}{9}\\[/tex]
13.
[tex]log_2{20}-log_{20}2=\\\\(log_2{2}+log_2{10})-(log_{20}20-log_{20}10)=\\\\(1+log_2{10})-(1-log_{20}10)=\\\\(1+\frac{log10}{log2})-(1-\frac{log10}{log20})=\\\\(1+\frac{1}{0,3})-(1-\frac{1}{log10+log2})=\\\\4\frac{1}{3}-(1-\frac{1}{1+0,3})=\\\\4\frac{1}{3}-\frac{3}{13}=\\\\\frac{160}{39}\\[/tex]
14.
[tex]2log(5x)^2+\frac{1}{2}logx-3log\frac{x}{2}=\\\\log(5x)^4+logx^{\frac{1}{2}}-log(\frac{x}{2})^3=\\\\log\frac{(5x)^4(x)^{\frac{1}{2} }}{(\frac{x}{2})^3}\\[/tex]
15.
[tex]log_{2}\frac{n}{3}-log_{2}3+log_{2}n^2\\\\log_{2}\frac{\frac{n}{3}}{3}*n^2\\\\log_{2}\frac{n^3}{9}\\[/tex]