Inverse, Exponential, and Logarithmic Functions 1.3
Review of exponent rules: b xb y  b x y ,
bx
1
 b x  y , x  b  x , (b x ) y  bxy , bx  0 for all x.
y
b
b
Properties of exponential functions: f(x) = bx
Domain (-∞,∞), Range (0,∞)
For all b>0 b0=1, so f(0)=1
If b>1, then the function is increasing. If 0<b<1, then the function is decreasing.
f(x) = ex is the natural exponential function. e≈2.718281828459…
Inverse Functions:
If a function has an inverse, then all of its x and y values have been switched on the inverse.
Ex. (2,4) inverse contains (4,2) and the function and its inverse together are symmetric to the
line y=x.
To determine if a function has an inverse, it must pass the horizontal line test.
Inverse
no inverse
*Functions that have inverses are one-to-one.
*If functions are inverses of each other, then f(g(x))=g(f(x)) = x.
To find the inverse of a function:
1) switch x and y
2) solve for y
Ex. Determine the intervals on which the functions have an inverse.
f(x)= |2x+1|
ex. Find the inverse of the function. Verify the relationships f(f-1(x)) = f-1(f(x)) = x. graph f and f-1
f(x) = x2+4 for x≥0
Logarithms:
Logarithms are the inverses of exponential functions. A logarithm is another way to represent
an exponent.
Special logs: Common log (base 10)
log 10 x = log x
Natural log(base e)
log e x = ln x
Shortcuts: ln ex = x, e lnx = x, log b bx = x, b log b x = x
Properties of exponents correspond to properties of logarithms.
Product Rule: logb(MN) = logb M + logb N
Quotient Rule: logb M = logb M – logbN
N
p
Power Rule: logb M = p logb M
(M,N and P are positive reals
and b≠1)
Change of Base Property: log b M = log M ex. evaluate log7 50 using natural log.
log b
x
xlnb
x+4
b =e , for all x ex. Express 2 using as an exponential function with base e.
Solving exponential and log functions.
To solve logarithmic equations: 1) Write as a single log. 2) Change the equation to exponential
form. 3) Solve the equation.
*Special case: If you have an equation in the form
and solve.
log b M = log b N, then you can set M = N
**You must check all answers for extraneous solutions. You CAN NOT take the log of a
negative or the log of zero.
Ex. log5x = -1
To solve exponential equations: 1) If you have only one variable, get it’s term by itself. 2) take
the log or ln of both sides. 3) Solve for x. or use same base properties.
Ex. 3x 5 x 5 
2
1
3
Ex. 53 x  29