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example 6 Solution of Exponential Equations
Chapter 5.3
Solve the following exponential equations:
a.
b.
24096 8 x
3 26 4 120x
2009 PBLPathways
2009 PBLPathways
Solve the following exponential equations:
a.
b.
24096 8 x
3 26 4 120x
2009 PBLPathways
Solve the following exponential equations:
a.
b.
24096 8 x
3 26 4 120x
Solving Exponential Equations Using Logarithmic Properties
To solve an exponential equation using logarithmic properties:
1.Rewrite the equation with a base raised to a power on one side.2.Take the logarithm, base e or 10, of both sides of the equation.3.Use a logarithmic property to remove the variable from the exponent.4.Solve for the variable.
Solving Exponential Equations Using Logarithmic Properties
To solve an exponential equation using logarithmic properties:
1.Rewrite the equation with a base raised to a power on one side.2.Take the logarithm, base e or 10, of both sides of the equation.3.Use a logarithmic property to remove the variable from the exponent.4.Solve for the variable.
2009 PBLPathways
Solve the following exponential equations:
a. 24096 8 x
2009 PBLPathways
Solve the following exponential equations:
a. 24096 8 x
1. Rewrite the equation with a base raised to a power on one side.
2
2
4096 8
log 4096 log 8
log 4096 2 log 8
x
x
x
2log 8 2l
log 4096 2 log 8
log 4096
2log 8
g 8
2
o
x
x
x
2009 PBLPathways
Solve the following exponential equations:
a. 24096 8 x
2. Take the logarithm, base e or 10, of both sides of the equation.
2
2
4096 8
log 4096 log 8
log 4096 2 log 8
x
x
x
2log 8 2l
log 4096 2 log 8
log 4096
2log 8
g 8
2
o
x
x
x
2009 PBLPathways
Solve the following exponential equations:
a. 24096 8 x
3. Use a logarithmic property to remove the variable from the exponent.
2
2
4096 8
log 4096 log 8
log 4096 2 log 8
x
x
x
2log 8 2l
log 4096 2 log 8
log 4096
2log 8
g 8
2
o
x
x
x
log logkM k M
2009 PBLPathways
Solve the following exponential equations:
a. 24096 8 x
4. Solve for the variable.
2
2
4096 8
log 4096 log 8
log 4096 2 log 8
x
x
x
2log 8 2l
log 4096 2 log 8
log 4096
2log 8
g 8
2
o
x
x
x
2009 PBLPathways
Solve the following exponential equations:
a. 24096 8 x
4. Solve for the variable.
2
2
4096 8
log 4096 log 8
log 4096 2 log 8
x
x
x
2log 8 2l
log 4096 2 log 8
log 4096
2log 8
g 8
2
o
x
x
x
2009 PBLPathways
Solve the following exponential equations:
a. 24096 8 x
4. Solve for the variable.
2
2
4096 8
log 4096 log 8
log 4096 2 log 8
x
x
x
2log 8 2l
log 4096 2 log 8
log 4096
2log 8
g 8
2
o
x
x
x
2009 PBLPathways
Solve the following exponential equations:
b. 3 26 4 120x
2009 PBLPathways
Solve the following exponential equations:
1. Rewrite the equation with a base raised to a power on one side.
b. 3 26 4 120x
3 2
3 2
3 2
6 4 120
4 20
ln 4 ln 20
3 2 ln 4 ln 20
3 2 l
ln 4 ln
n 4 ln 20
4
x
x
x
x
x
ln 203 2
ln 4
ln 203 2
ln 4
ln 202
ln 41.387
3
x
x
x
2009 PBLPathways
Solve the following exponential equations:
1. Rewrite the equation with a base raised to a power on one side.
b. 3 26 4 120x
3 2
3 2
3 2
6 4 120
4 20
ln 4 ln 20
3 2 ln 4 ln 20
3 2 l
ln 4 ln
n 4 ln 20
4
x
x
x
x
x
ln 203 2
ln 4
ln 203 2
ln 4
ln 202
ln 41.387
3
x
x
x
2009 PBLPathways
Solve the following exponential equations:
2. Take the logarithm, base e or 10, of both sides of the equation.
b. 3 26 4 120x
3 2
3 2
3 2
6 4 120
4 20
ln 4 ln 20
3 2 ln 4 ln 20
3 2 l
ln 4 ln
n 4 ln 20
4
x
x
x
x
x
ln 203 2
ln 4
ln 203 2
ln 4
ln 202
ln 41.387
3
x
x
x
2009 PBLPathways
Solve the following exponential equations:
3. Use a logarithmic property to remove the variable from the exponent.
b. 3 26 4 120x
3 2
3 2
3 2
6 4 120
4 20
ln 4 ln 20
3 2 ln 4 ln 20
3 2 l
ln 4 ln
n 4 ln 20
4
x
x
x
x
x
ln 203 2
ln 4
ln 203 2
ln 4
ln 202
ln 41.387
3
x
x
x
log logkM k M
2009 PBLPathways
Solve the following exponential equations:
4. Solve for the variable.
b. 3 26 4 120x
3 2
3 2
3 2
6 4 120
4 20
ln 4 ln 20
3 2 ln 4 ln 20
3 2 l
ln 4 ln
n 4 ln 20
4
x
x
x
x
x
ln 203 2
ln 4
ln 203 2
ln 4
ln 202
ln 41.387
3
x
x
x
2009 PBLPathways
Solve the following exponential equations:
4. Solve for the variable.
b. 3 26 4 120x
3 2
3 2
3 2
6 4 120
4 20
ln 4 ln 20
3 2 ln 4 ln 20
3 2 l
ln 4 ln
n 4 ln 20
4
x
x
x
x
x
ln 203 2
ln 4
ln 203 2
ln 4
ln 202
ln 41.387
3
x
x
x
2009 PBLPathways
Solve the following exponential equations:
4. Solve for the variable.
b. 3 26 4 120x
3 2
3 2
3 2
6 4 120
4 20
ln 4 ln 20
3 2 ln 4 ln 20
3 2 l
ln 4 ln
n 4 ln 20
4
x
x
x
x
x
ln 203 2
ln 4
ln 203 2
ln 4
ln 202
ln 41.387
3
x
x
x
2009 PBLPathways
Solve the following exponential equations:
4. Solve for the variable.
b. 3 26 4 120x
3 2
3 2
3 2
6 4 120
4 20
ln 4 ln 20
3 2 ln 4 ln 20
3 2 l
ln 4 ln
n 4 ln 20
4
x
x
x
x
x
ln 203 2
ln 4
ln 203 2
ln 4
ln 202
ln 41.387
3
x
x
x
2009 PBLPathways
Solve the following exponential equations:
4. Solve for the variable.
b. 3 26 4 120x
3 2
3 2
3 2
6 4 120
4 20
ln 4 ln 20
3 2 ln 4 ln 20
3 2 l
ln 4 ln
n 4 ln 20
4
x
x
x
x
x
ln 203 2
ln 4
ln 203 2
ln 4
ln 202
ln 41.387
3
x
x
x
2009 PBLPathways
Solve the following exponential equations:
Alternate strategy.
b. 3 26 4 120x
ln 202
ln 41.387
3x
3 2
3 2
4
4
4
6 4 120
4 20
3 2 log 20
3 log 20 2
log 20 2
3
x
x
x
x
x
2009 PBLPathways
Solve the following exponential equations:
Alternate strategy.
b. 3 26 4 120x
ln 202
ln 41.387
3x
3 2
3 2
4
4
4
6 4 120
4 20
3 2 log 20
3 log 20 2
log 20 2
3
x
x
x
x
x
2009 PBLPathways
Solve the following exponential equations:
Alternate strategy.
b. 3 26 4 120x
ln 202
ln 41.387
3x
3 2
3 2
4
4
4
6 4 120
4 20
3 2 log 20
3 log 20 2
log 20 2
3
x
x
x
x
x
2009 PBLPathways
Solve the following exponential equations:
Alternate strategy.
b. 3 26 4 120x
ln 202
ln 41.387
3x
3 2
3 2
4
4
4
6 4 120
4 20
3 2 log 20
3 log 20 2
log 20 2
3
x
x
x
x
x
2009 PBLPathways
Solve the following exponential equations:
Alternate strategy.
b. 3 26 4 120x
ln 202
ln 41.387
3x
3 2
3 2
4
4
4
6 4 120
4 20
3 2 log 20
3 log 20 2
log 20 2
3
x
x
x
x
x
2009 PBLPathways
Solve the following exponential equations:
Alternate strategy.
b. 3 26 4 120x
ln 202
ln 41.387
3x
3 2
3 2
4
4
4
6 4 120
4 20
3 2 log 20
3 log 20 2
log 20 2
3
x
x
x
x
x
