`log_3(x^2) = 4.5` Solve the equation accurate to three decimal places

To solve a logarithmic equation, we may simplify  or rewrite it using the properties of logarithm.

For the given problem `log_(3)(x^2)=4.5` ,  we may apply the property:

`a^((log_(a)(x))) = x`

 The "log" cancels out which we need to accomplish on the left side of the equation.

Raising both sides by the base of 3:

`3^((log_3(x^2))) = 3^(4.5)`

 ` x^2= 3^(4.5) `

Taking the square root on both sides:

`sqrt(x^2) =+-sqrt(3^(4.5))`

`x= +-11.84466612`

Rounded off to three decimal places:

`x=+-11.845` .

Plug-in the x-values to check if they are the real solution:

`log_3(11.845^2)=4.5 ` so x = 11.845 is a real solution.

Now let `x=-11.845` 

 `log_3((-11.845)^2)`

`log_3(140.304025)=4.5` so x = -11.845 is a real solution.

So, x= 11.845, x = -11.845 are both solutions.