Math 201
Exponentials and logarithms Solutions
The purpose of this worksheet is for you to work through some of the basic properties of the
exponential and logarithmic functions. Recall that an exponential function is of the form P(t) =
P0atwhere P0is the “initial quantity” and ais the “base”.
(1) Use you calculator to graph P(t)=2tand Q(t) = .5tand sketch these graphs.
In general if a > 1 the exponential function will increase, this is called “exponential growth”.
If 0 <a<1 then the function will decrease, this is called “exponential decay”.
(2) Recall that the inverse function to fis denoted f−1and is defined by the property
f−1(y) = xexactly when f(x) = y.
What does this tell us about (f◦f−1)(y)? What about (f−1◦f)(x)?
(f◦f−1)(y) = f(f−1(y)) = f(x) = y, So (f◦f−1)(y) = y.
(f−1◦f)(x) = f−1(f(x)) = f−1(y) = x, So (f−1◦f)(x) = x).
(3) Logarithms are defined to be inverse functions to exponential functions. The most com-
monly used bases are eand 10, therefore we will only discuss the logarithms associated with
these bases. First of all, what is e? Find this button on your calculator and write down the
first five digits. (In Math 202 you will learn how your calculator does this.)
e= 2.71828
(4) The inverse function to f(x) = exis f−1(x) = ln(x) and is read as “log base e” or “natural
log”. Use the compositions in (2) to derive two different rules about how exand ln(x)
interact.
x= (f◦f−1)(x) = f(f−1(x)) = f(ln(x)) = eln(x), So eln(x)=x
x= (f−1◦f)(x) = f−1(f(x)) = f−1(ex) = ln(ex), So ln(ex) = x
(5) Repeat the previous part for g(x) = 10xwith inverse g−1(x) = log(x), that is “log base 10”.
10log(x)=xand log(10x) = x
(6) Use you calculator to graph ln(x) and log(x). Sketch the graphs.
(7) Use (4) to solve the equation 17 = 2(ex+1)
17
2=ex+1
(1)
ln 17
2= ln(ex+1)(2)
ln 17
2=x+ 1(3)
ln 17
2−1 = x(4)
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