Question

Pls, can someone explain me the solution step by step :)​

Answer 1

4

Explanation:

Given expression:

`\left(2-\log_(√(2))10\right)\left(2-\log_(√(5))10\right)`

`\textsf{Apply the radical rule}: \quad √(a)=a^{(1)/(2)}`

`\implies \left(2-\log_{2^{(1)/(2)}}10\right)\left(2-\log_{5^{(1)/(2)}}10\right)`

`\textsf{Apply the log rule}: \quad \log _(a^n)(x)=(1)/(n)\log _a\left(x\right),\:\quad \:a > 0`

`\implies \left(2-(1)/((1)/(2))\log_(2)10\right)\left(2-(1)/((1)/(2))\log_(5)10\right)`

`\implies \left(2-2 \log_(2)10\right)\left(2-2\log_(5)10\right)`

Rewrite 10 as 2 · 5 and 10 as 5 · 2:

`\implies \left(2-2 \log_(2)(2 \cdot 5)\right)\left(2-2\log_(5)(5 \cdot 2)\right)`

`\textsf{Apply the log product law}: \quad \log_axy=\log_ax + \log_ay`

`\implies \left(2-(2 \log_(2)2+2\log_2 5)\right)\left(2-(2\log_(5)5 +2\log_52)\right)`

`\implies \left(2-2 \log_(2)2-2\log_2 5\right)\left(2-2\log_(5)5 -2\log_52\right)`

`\textsf{Apply log law}: \quad \log_aa=1`

`\implies \left(2-2(1)-2\log_2 5\right)\left(2-2(1) -2\log_52\right)`

`\implies \left(2-2-2\log_2 5\right)\left(2-2 -2\log_52\right)`

`\implies \left(-2\log_2 5\right)\left(-2\log_52\right)`

Multiply:

`\implies 4 \cdot \log_2 5 \cdot \log_52`

`\textsf{Apply the log rule}: \quad \log _(a)(x) \cdot \log_(x)(a)=1`

`\implies 4 \cdot 1`

`\implies 4`