0000:00:00:00:00Start20Number 20ExplanationIf x1x_1x1 and x2x_2x2 satisfy the equation (2logx−1)⋅1logx10=log10(2 \log x - 1) \cdot \frac{1}{\log_x 10} = \log 10(2logx−1)⋅logx101=log10, then x1x2=....x_1x_2 = ....x1x2=....5105\sqrt{10}5104104\sqrt{10}4103103\sqrt{10}3102102\sqrt{10}21010\sqrt{10}10ExplanationLet p=log10x=logxp = \log_{10} x = \log xp=log10x=logx, then (2logx−1)⋅1logx10=log10(2 \log x - 1) \cdot \frac{1}{\log_x 10} = \log 10(2logx−1)⋅logx101=log10(2log10x−1)⋅log10x=1(2 \log_{10} x - 1) \cdot \log_{10} x = 1(2log10x−1)⋅log10x=1(2p−1)p=1(2p - 1)p = 1(2p−1)p=12p2−p−1=02p^2 - p - 1 = 02p2−p−1=0(2p+1)(p−1)=0(2p + 1)(p - 1) = 0(2p+1)(p−1)=0p=−12∪p=1p = -\frac{1}{2} \cup p = 1p=−21∪p=1 Determining the value of xxx from ppp p=−12→log10x=−12→x1=10−12p = -\frac{1}{2} \rightarrow \log_{10} x = -\frac{1}{2} \rightarrow x_1 = 10^{-\frac{1}{2}}p=−21→log10x=−21→x1=10−21p=1→log10x=1→x2=10p = 1 \rightarrow \log_{10} x = 1 \rightarrow x_2 = 10p=1→log10x=1→x2=10 So the product of x1x2x_1x_2x1x2 10−12⋅10=1012=1010^{-\frac{1}{2}} \cdot 10 = 10^{\frac{1}{2}} = \sqrt{10}10−21⋅10=1021=10
20Number 20ExplanationIf x1x_1x1 and x2x_2x2 satisfy the equation (2logx−1)⋅1logx10=log10(2 \log x - 1) \cdot \frac{1}{\log_x 10} = \log 10(2logx−1)⋅logx101=log10, then x1x2=....x_1x_2 = ....x1x2=....5105\sqrt{10}5104104\sqrt{10}4103103\sqrt{10}3102102\sqrt{10}21010\sqrt{10}10ExplanationLet p=log10x=logxp = \log_{10} x = \log xp=log10x=logx, then (2logx−1)⋅1logx10=log10(2 \log x - 1) \cdot \frac{1}{\log_x 10} = \log 10(2logx−1)⋅logx101=log10(2log10x−1)⋅log10x=1(2 \log_{10} x - 1) \cdot \log_{10} x = 1(2log10x−1)⋅log10x=1(2p−1)p=1(2p - 1)p = 1(2p−1)p=12p2−p−1=02p^2 - p - 1 = 02p2−p−1=0(2p+1)(p−1)=0(2p + 1)(p - 1) = 0(2p+1)(p−1)=0p=−12∪p=1p = -\frac{1}{2} \cup p = 1p=−21∪p=1 Determining the value of xxx from ppp p=−12→log10x=−12→x1=10−12p = -\frac{1}{2} \rightarrow \log_{10} x = -\frac{1}{2} \rightarrow x_1 = 10^{-\frac{1}{2}}p=−21→log10x=−21→x1=10−21p=1→log10x=1→x2=10p = 1 \rightarrow \log_{10} x = 1 \rightarrow x_2 = 10p=1→log10x=1→x2=10 So the product of x1x2x_1x_2x1x2 10−12⋅10=1012=1010^{-\frac{1}{2}} \cdot 10 = 10^{\frac{1}{2}} = \sqrt{10}10−21⋅10=1021=10
