SOLUTION: Find the range and domain of the following logarithmic function: log_(2) sqrt(2x-3)

Algebra ->  Exponential-and-logarithmic-functions -> SOLUTION: Find the range and domain of the following logarithmic function: log_(2) sqrt(2x-3)       Log On


   

Question 1103714: Find the range and domain of the following logarithmic function: log_(2) sqrt(2x-3)
Found 2 solutions by Edwin McCravy, Theo:
Answer by Edwin McCravy(20060) About Me  (Show Source):
You can put this solution on YOUR website!
Find the range and domain of the following logarithmic function:
log%282%2Csqrt%282x-3%29%29
We can only take logarithms of positive numbers, therefore

2x-3 must be positive, or

2x-3 > 0
  2x > 3
   x > 3%2F2

So the domain is %28matrix%281%2C3%2C3%2F2%2C%22%2C%22%2Cinfinity%29%29

Since sqrt%282x-3%29 takes on all positive values,
the range is %28matrix%281%2C3%2C-infinity%2C%22%2C%22%2Cinfinity%29%29

Edwin

Answer by Theo(13342) About Me  (Show Source):
You can put this solution on YOUR website!
log of any expression will only give you a real answer when the expression is greater than 0.

this means that 2x-3 must be greater than 0.

in the expression 2x-3, solve for x to get x > 3/2.

the function will give you a real value of y only if the value of x is greater than 3/2.

your domain is all real values of x such that x > 3/2.

by definition of logs, y = logb(x) if and only if b^y = x.

your equation is y = log2(2x-3).

this is true if and only if 2^y = 2x-3.

since 2x-3 must be greater than 0, then 2^y = 2x-3 is equivalent to 2^y > 0.

2^y will always be greater than 0 for all real values of y.

since 2^y will always be greater than 0, regardless if y is itself > 0 or = 0 or < 0, then your range is all real values of y.

if you graph the equation of y = log2(3x-2), it will look like this:

$$$

as the value of x approaches 1.5 from the right on the graph, the value of y becomes more and more negative.

your domain is all values of x such that x >= 3/2.

your range is all real values of y.