Question

729^(2x-5) = 9 Write answer a/b form in simplified form

Answer 1

Given the expression:

`\text{ 729}^{2x\text{ - 5}}\text{ = 9}`

Let's simplify the expression to find x.

`\text{ 729}^{2x\text{ - 5}}\text{ = 9}`
`\log _(10)(729^(2x-5))=\log _(10)(9)`
`(2x-5_{})\log _(10)(729)=\text{ }\log _(10)(9)`
`\text{ 2x - 5 = }(\log_(10)(9))/(\log_(10)(729))`
`\text{ 2x - 5 + 5 = }\log _(729)(9)\text{ + 5 ; Log Rule (}(\log_c\left(b\right))/(\log_c\left(a\right))=\log _a\mleft(b\mright))`
`\text{ 2x = }\log _(3^6)(9)\text{ + 5}`
`\text{ 2x = }(1)/(6)\log _3(9)\text{ + 5 ; Log Rule }(\log _(a^b)\mleft(x\mright)=(1)/(b)\log _a\mleft(x\mright))`
`\text{ 2x = }(1)/(6)\log _3(3^2)\text{ + 5}`
`\text{ 2x = (}(1)/(6))(2)\text{ + 5 ; Log Rule (}\log _a\mleft(a^x\mright)=x)`
`\text{ 2x = }(2)/(6)+5\text{ }\rightarrow\text{ 2x = }(2)/(6)\text{ + }(30)/(6)`
`\text{ 2x = }(32)/(6)\text{ }\rightarrow\text{ (2x)(}(1)/(2))\text{ = (}(32)/(6))((1)/(2))`
`\text{ x = }(32)/(12)`
`\text{ x = }((32)/(4))/((12)/(4))=\text{ }(8)/(3)`

Therefore, x = 8/3.