To find the domain of the given logarithmic function, we will recall the definition of a logarithm.
log_b x=y ⇔ x= b^y
This tells us how we can rewrite the logarithm equivalent to y as an exponential equation. 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>0
Furthermore, 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 exponential function. Since graphing an exponential function is easier than graphing a logarithmic function, we will start by calculating the inverse of the given function.
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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.
