Logarithm Worksheet With Answer Key Page 2
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4Log Review (5.13)
Logarithms as Inverses
Through these problems you will see the logarithm functions are the inverses of the
exponential functions.
Exercises
1. Consider the exponential function f(x) = 2
x
.
a. Graph f(x) on the grid at right.
Explain why f(x) has an inverse that is a function.
−1
b. Draw the graph of f
(y)
(keeping in mind that the meaning of
the axes changes depending on
which relationship you are looking at)
−1
c. Algebraically find f
(y).
=
⋅
+
Example 1 : Find the inverse of the function
(
)
4
3
1
. Graph f and its inverse to
x
f
x
confirm your inverse visually.
=
⋅
+
(
)
4
3
1
①
The original function
x
f
x
=
⋅
+
4
3
1
②
We are calling the output of f(x) “y”
x
y
y −
=
(
1) / 4 3
x
④
Solve for x
−
⎛
⎞
(
1)
y
= =
−
1
log
( )
⎜
⎟
x
f
y
3
4
⎝
⎠
−
⎛
⎞
(
1)
y
−
=
1
Since
log
( )
can not be graphed, use the Change of Base formula to
⎜
⎟
f
y
3
⎝
4
⎠
−
(
1)
log
y
=
−
1
convert to a form that can be graphed, namely,
4
( )
. Also, your “Y=” menu
f
y
log 3
uses X as its input, so you will enter the expression as “Y
=log((X-1)/4)/log(3)”. Also
2
note that “ZOOM-Square” will give you a better picture (why?).
Since the two functions look symmetric over y = x, We
can be very confident the inverse was made correctly.
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