Let's begin by finding the and . Then, we will graph the given .
Finding the Domain and Range
To find the domain of the given logarithmic function, we will recall the definition of a .
log_b x=y ⇔ x= b^y
This tells us how we can rewrite the logarithm equivalent to y as an . The argument x is equal to the base b raised to the power of y. The base b is defined as a positive number different than one. Since b>0 and b≠1, it follows that b^y is also greater than 0.
b>0 ⇒ b^y>0Furthermore, because x= b^y, x is also greater than 0. Therefore, in the expression log_b x, we have that x>0. Let's now consider the given function.
y=log_5 ( x-1)
We know that x-1>0, which implies x>1. This is the domain. The range of a logarithmic function is always all real numbers.
Domain:& x>1
Range:& All real numbers
Graphing the Function
Recall that the inverse of a logarithmic function is an . Since graphing an exponential function is easier than graphing a logarithmic function, we will start by calculating the inverse of the given function.
ccc
Function & & Inverse
y=log_5 (x-1) & & y=5^x+1
Now, we will graph the inverse function and reflect the graph of the inverse function across the line y=x.
Graphing the Inverse Function
To draw the graph of y=5^x+1, we will construct a table of values.
| x
|
5^x+1
|
y=5^x+1
|
| - 1
|
5^(- 1)+1
|
1.2
|
| 0
|
5^0+1
|
2
|
| 1
|
5^1+1
|
6
|
| 2
|
5^2+1
|
26
|
Let's plot the points ( - 1, 1.2), ( 0, 2), ( 1, 6), and ( 2, 26), and connect them with a smooth curve.
Reflecting the Inverse Function Across the Line y=x
To reflect the curve across the line y=x, we can reverse the coordinates of the points we found. Then we will plot these new points and connect them with a smooth curve.
Finally, let's remove the line y=x and the graph of the inverse function.
