Objectives
Exponential and
Logarithmic Equations
– Use like bases to solve exponential equations.
– Use logarithms to solve exponential equations.
– Use the definition of a logarithm to solve
logarithmic equations.
– Use the one-to-one property of logarithms to solve
logarithmic equations.
Solving Equations
Solving Exponential Equations
• Use the properties we have learned about
exponential & logarithmic expressions to
solve equations that have these expressions
in them.
• Equations with variables in the exponents, such
as 3x = 40 and 53x = 25, are called exponential
equations.
• For exponential equations, if the base is the
same on both sides of the equation, the
exponents must also be the same (equal!)
• Find values of x that will make the
logarithmic or exponential equation true.
bM = b N , M = N
Example
• Solve:
52 x −3 = 125
5 2 x−3 = 53
2x − 3 = 3
2x = 6
x=3
1
Example
log2x = log50
xlog2 = log50
x=
Sometimes it is easier to solve a
logarithmic equation than an
exponential one.
• Any exponential equation can be
rewritten as a logarithmic one, then you
can apply the properties of logarithms.
log50
log2
Example
• Solve:
5
2x −1
= 99
Example
Solve: e−0.25w = 12
Solve
32x −1 = 5 x +1
2
Solving Logarithmic Equations
• Equations containing variables in logarithmic
expressions, such as log2 x = 16 and
log x + log (x + 4) = 1, are called logarithmic
equations.
• To solve logarithmic equations algebraically,
we first try to obtain a single logarithmic
expression on one side and then write an
equivalent exponential equation.
Example
Solve:
Example
log4 x = −3
Solve:
log4 (x + 3) − log4 (5x − 1) = 3
log 4 641 ? − 3
log 4 4−3 ? − 3
− 3 = −3
1
.
64
Example
• Solve: log 2 ( x + 1) + log 2 ( x − 1) = 3
Check Solution to Example
Check: For x = 3: log 2 (3 + 1) + log 2 (3 − 1) = 3 →
log 2 4 + log 2 2 = ? 3
log 2 (4 ⋅ 2) = ? 3
log 2 8 = ? 3
3=3
For x = −3: log 2 (−3 + 1) + log 2 (−3 − 1)? 3
log 2 (−2) + log 2 (−4)? 3
Negative numbers do not have real-number
logarithms. The solution is 3.
3
Example
• Solve:
ln x − ln( x − 4) = ln 3
4