Question

Pls Help, Max Points

Answer 1

`\\ \rm\Rrightarrow 36^x-7(6^x)-18=0`

`\\ \rm\Rrightarrow 6^(2x)-7(6^x)-18=0`

`\\ \rm\Rrightarrow (6^x)^2-7(6^x)-18=0`

• Let 6^x=p

`\\ \rm\Rrightarrow p²-7p-18=0`

`\\ \rm\Rrightarrow p²-9p+2p-18=0`

`\\ \rm\Rrightarrow p(p-9)+2(p-9)=0`

`\\ \rm\Rrightarrow (p+2)(p-9)=0`

Omit negative as we have to solve for real nos

• p=9

put value

`\\ \rm\Rrightarrow 6^x=9`

`\\ \rm\Rrightarrow ln6^x=ln3^2`

`\\ \rm\Rrightarrow xln6=2ln3`

`\\ \rm\Rrightarrow x=(2ln3)/(ln6)`

Answer 2

`\large \text{$ x = (2\ln 3)/(\ln 6) $}`

Explanation:

`\large \begin{aligned}36^x-7(6^x)-18 & =0\\(6^2)^x-7(6^x)-18 & =0\\(6^x)^2-7(6^x)-18 & =0\\\\\textsf{let }6^x=y\implies y^2-7y-18 & = 0\\(y-9)(y+2) & = 0\\\implies y & = 9, -2\\\\\implies 6^x & = 9, -2\\\textsf{Cannot take logs of -ve numbers} \implies \ln 6^x & = \ln 9 \quad \sf(only)\\x \ln 6 & = 2\ln 3\\x & = (2\ln 3)/(\ln 6)\\\end{aligned}`