Solve this equation: `ln(-2x)=lnx^2` for logarithms.



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Posted on (Answer #1)


To eliminate the logarithm, apply the rule `e^(lnx)=x` .



Express the equation in quadratic form `ax^2+bx+c=0` .


Then, factor.


Set each factor to zero and solve for x.

`x+2=0`               and           `x=0`


Then, substitute the values of x to the original equation to check.

`x=-2` ,    `ln (-2*-2) = ln(-2)^2`

                                 `ln4 = ln4`           (True)

`x=0 ` ,           `ln(-2*0)= ln(0)^2`

                                 `ln 0 = ln 0`            (Invalid logarithm)

Note that in logarithm, we cannot have a zero as the argument. It should always be greater than zero.

Hence, the solution to the equation `ln(-2x)=lnx^2`  is  `x=-2` .


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