Transcript Section 3.4 - Shelton State Community College

Section 3.4
Exponential and Logarithmic
Equations
Overview
• In this section we will solve logarithmic and
exponential equations.
• Some things we will need:
1. The ability to convert from exponential to logarithmic
and vice versa.
2. The properties of logarithms.
3. The change of base formula.
4. The ability to solve linear and quadratic equations
5. Knowledge of the domain of a logarithmic function.
A Couple of (New) Things
1. The exponential function is one-to-one:
If ax = ay, then x = y.
2. The logarithmic function is one-to-one. In
both directions:
If x = y, then logax = logay.
If logax = logay, then x = y.
Case I: Exponential
• Exponential equations do not have the word
“log” anywhere in the problem.
• To solve an exponential equation:
1. Write both sides of the equation as powers of the
same base. Then set the exponents equal to each
other.
2. Take the natural log of both sides (Know the
difference between an exact answer and an
approximate answer).
Expressing Your Answers
• An exact answer will leave the log expressions
intact. No decimal approximations will be
used for any log expressions.
• An approximate answer will involve using a
scientific calculator to find approximate values
for log expressions. Answers will be rounded
to a designated number of decimal places.
Examples
8 2 x 1  512
5
e
x 7
4
x2
 5
1

e
6 4 x 1  5 x 3
Case II: Log = number
• These equations will have a log expression on
one side and a number on the other.
• Solve by converting to exponential form:
logax = y is the same as x = ay
Examples
log3 x  4
ln x  8
6 ln 2 x   18
Case III: Multiple logs = number
• These equations will have more than one log
expression on one side and a number on the
other side.
• Use the properties of logarithms to combine
the multiple logs into a single logarithmic
expression.
• Then convert to exponential.
Case IV: Logs on both sides
• If you have multiple logs on either side, use
the properties of logarithms to condense
them into single logarithmic expression.
• Then set the arguments equal to each other
and solve.
Examples
log2 ( x  3)  log2 ( x  1)  5
logx  6  log x  log 6
3 log x  log125
2 log x  log 9  log 576