`log_5sqrt(x - 4) = 3.2` Solve the equation accurate to three decimal places

To simplify the logarithmic equation: `log_5(sqrt(x-4))=3.2` , recall the logarithm property:  `a^((log_(a)(x))) = x` .

When a logarithm function is raised by the same base, the log cancels out which is what we need to do on the left side of the equation.

As a rule we apply same change on both sides of the equation.

 Raising both sides by base of 5:

`5^(log_5(sqrt(x-4)))= 5^(3.2)`

`sqrt(x-4) = 5^(3.2)`

To cancel out the radical sign, square both sides:

`(sqrt(x-4))^2 = (5^(3.2)) ^2 `  

  `x-4 =5^(6.4)`

`  x= 5^(6.4)+4`

 ` x~~29748.593`  (rounded off to three decimal places)

To check, plug-in `x=29748.593` in `log_5(sqrt(x-4))` :

`log_5(sqrt(29748.593-4))`

`log_5(sqrt(29744.593))`

`log_5(172.4662083)=3.2` which is what we want

So, x=29748.593 is the real solution.

Note:` (x^m)^n= x^((m*n ))`