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=4log_(10) x+5
We know that x>0. This is the domain. The range of a logarithmic function is always all real numbers.
Domain:& x>0
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=4logx+5 & & y=10^(x-54)
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=10^(x-54), we will construct a table of values.
x
10^(x-54)
y=10^(x-54)
- 3
10^(- 3-54)
0.01
1
10^(1-54)
0.1
5
10^(5-54)
1
9
10^(9-54)
10
Let's plot the points ( - 3, 0.01), ( 1, 0.1), ( 5, 1), and ( 9, 10), 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.
