To evaluate the given equation `10^(3x-10)=(1/100)^(6x-1)` , we may apply `100=10^2` . The equation becomes:
`10^(3x-10)=(1/10^2)^(6x-1)`
Apply Law of Exponents: `1/x^n = x^(-n)` .
`10^(3x-10)=(10^(-2))^(6x-1)`
Note:` 1/100= 10^(-2)`
Apply Law of Exponents: `(x^n)^m = x^(n*m)` .
`10^(3x-10)=10^((-2)*(6x-1))`
`10^(3x-10)=10^(-12x+2)`
Apply the theorem: If `b^x=b^y` then `x=y` , we get:
`3x-10=-12x+2`
Add `12x` on both sides of the equation.
`3x-10+12x=-12x+2+12x`
`15x-10=2`
Add `10` on both sides of the equation.
`15x-10+10=2+10`
`15x=12`
Divide both sides by `15` .
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To evaluate the given equation `10^(3x-10)=(1/100)^(6x-1)` , we may apply `100=10^2` . The equation becomes:
`10^(3x-10)=(1/10^2)^(6x-1)`
Apply Law of Exponents: `1/x^n = x^(-n)` .
`10^(3x-10)=(10^(-2))^(6x-1)`
Note:` 1/100= 10^(-2)`
Apply Law of Exponents: `(x^n)^m = x^(n*m)` .
`10^(3x-10)=10^((-2)*(6x-1))`
`10^(3x-10)=10^(-12x+2)`
Apply the theorem: If `b^x=b^y` then `x=y` , we get:
`3x-10=-12x+2`
Add `12x` on both sides of the equation.
`3x-10+12x=-12x+2+12x`
`15x-10=2`
Add `10` on both sides of the equation.
`15x-10+10=2+10`
`15x=12`
Divide both sides by `15` .
`(15x)/15=12/15`
`x=12/15`
Simplify.
`x=4/5`
Checking: Plug-in `x=4/5` on `10^(3x-10)=(1/100)^(6x-1)` .
`10^(3*(4/5)-10)=?(1/100)^(6*(4/5)-1)`
`10^(12/5-10)=?(1/100)^(24/5-1)`
`10^(12/5-50/5)=?(1/100)^(24/5-5/5)`
`10^((-38)/5)=?(1/100)^(19/5)`
`10^((-38)/5)=?(10^(-2))^(19/5)`
`10^((-38)/5)=?10^((-2)*19/5)`
`10^((-38)/5)=10^((-38)/5) ` TRUE
Thus, the `x=4/5` is the real exact solution of the equation `10^(3x-10)=(1/100)^(6x-1)` .
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