Exponential and Logarithmic E Equations ti
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Solving Exponential Equations Terms that have a base on one side and a power of that base on the other ...
Solving Exponential Equations Terms that have a base on one side and a power of that base on the other side can be solved using the property:
am = an ⇔ m = n
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Example 2−3 = 2 x
Solve 2
2 1 = 2 x −4 x 8
−4 x
−3 = x 2 − 4 x
x2 − 4x + 3 = 0
( x − 3)( x − 1) = 0 x = 1, x = 3
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Practice
Solve 25 = 5x+3
52 = 5 x+3 2 = x+3 x = −1
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Solving Exponential Equations Watch for quadratic forms where we have
b2 x = ( b x )
2
We can often factor expressions containing terms like
b 2 x and b x
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Example Write in quadratic form Factor Use Zero Property Solve for ex
e2 x − 2e x − 8 = 0
( e ) − 2e ( e − 4 )( e x 2
x
−8 = 0
x
x
+ 2) = 0
e x − 4 = 0 or e x + 2 = 0 e x = 4 or e x = −2
Recall R(ex) >0
ex = 4
Take the ln of each side
x = ln 4 6
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Solving Exponential Equations General Guidelines – Step 1: Isolate the exponential expression Put your exponential expression on one side everything outside of the exponential expression on the other side of your equation Step 2: Take the natural log of both sides The inverse operation of an exponential expression is a log. Make sure that you do the same thing to both sides of your equation to keep them equal to each other 7
Solving Exponential Equations Step 3: Use the properties of logs to pull the x out of the exponent Step 4: Solve for x Now that the variable is out of the exponent, solve for the variable using inverse operations to complete the problem
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Example Isolate the exponent
5 (10 )
(10 )
x+1
=
x+1
= 12
12 5
Take the log of each side
x + 1 = log
Solve for x
x = log
12 5
12 −1 5
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3
e3 x = 50
Practice Take the ln of each side Solve for x
3x = ln 50 x=
ln 50 3
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Solving Logarithmic Equations of the Form log b x = y Step 1: Write as one log isolated on one side Get your log on one side everything outside of the log on the other side of your equation using inverse operations. Also use properties of logs to write it so that th there iis only l one log l Step 2: Use the definition of logarithms to write in exponential form A reminder that the definition of logarithms is the logarithmic function with base b, where b > 0 and b ≠ 0, and is defined as logbx = y if and only if by = x 11
Solving Logarithmic Equations of the Form log b x = y Step 3: Solve for x Now that the variable is out of the log, solve for the variable using inverse operations to complete the problem Step 4: Verify the domain This is necessary as the domain of log(x) is strictly positive reals
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Example
log3 ( x + 24) − log 3 ( x + 2) = 2
Use properties of log
⎛ x + 24 ⎞ log 3 ⎜ ⎟=2 ⎝ x+2 ⎠
Rewrite as exponent
x + 24 = 32 = 9 x+2
Cross multiply
x + 24 = 9 ( x + 2 )
Expand
x + 24 = 9 x + 18 3 x= 4
Solve and verify domain
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Example
3 + ln x − 2 = 4 ln x − 2 = 1 1 ln ( x − 2 ) = 1 2 ln ( x − 2 ) = 2
Subtract 3 Property of log Multiply by 2 Write in exponential form
x − 2 = e2
Add 2
x = 2 + e2
Verify domain
2 + e2 − 2 > 0
{2 + e } 2
Practice
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Solve for x 1 − log ( x − 2 ) = log ( 3x + 1) 1 = log ( x - 2 ) + log ( 3 x + 1)
log10 = 1 = log ⎡⎣( x - 2 )( 3 x + 1) ⎤⎦ 10 = ( x - 2 )( 3 x + 1) 10 = 3 x 2 − 5 x − 2 3 x 2 − 5 x − 12 = 0 ( 3x + 4 )( x − 3) = 0 4 x = − ,x =3 3
disgard negative solution and x = 3 15
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Solving Mixed Equations • Exponential or logarithmic equations that mix bases
3x + 5x = 12 • Equations that mix exponential and logarithmic expressions x
3 + ln l x=5
• Equations that mix transcendental and algebraic expressions
3x + 5 x 2 = 5
• Use Graphic Method to solve these mixed equations 16
Example
3x + 5x = 12
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Example
3x + ln x = 5
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Practice
3x + 5 x 2 = 5
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Remarks We often find it more useful to think in terms of the time required for money to double, instead of in interest rates and time compounded. It would be is simpler if you were told that anything invested will double in 7 years. years
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Find time to double, if the interest rate is 6% compounded quarterly
Find time to double, if the interest rate is 6% compounded quarterly
Example
Answer: 11.64 years 22
Remarks We can approximate the time to double using
t=
70 r%
Example: Find the approximate time to double, if the interest rate is 6% compounded quarterly 70 = 11.666… ≈ 11.67 years 6
t=
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Find time to triple, if the interest rate is 8.5% compounded continuously
Practice
We want to know when e0.085t = 3
Pe rt = Pe0.085t e0.085t = 3 0.085t = ln 3 t=
ln 3 0.085
t=
ln 3 0.085
Answer: 12.92 years
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Example
You invested in property near the River Walk in 2004 for $325,000 and sold it in 2008 for $685,000. What is the annual rate of return for this investment?
t = 2008 − 2004 = 4
n =1
$325, 000 (1 + r ) = 685, 000 4
(1 + r )
4
=
685, 000 137 = 65 325, 000
4 ⎛ 137 ⎞ ln (1 + r ) = ln ⎜ ⎟ ⎝ 65 ⎠
Example
1 ⎛ 137 ⎞ ln ⎜ ⎟ ⎝ 65 ⎠
⎛ r⎞ Using P ⎜1 + ⎟ ⎝ n⎠
nt
⎛ 137 ⎞ 4 ln (1 + r ) = ln ⎜ ⎟ ⎝ 65 ⎠
1 ⎛ 137 ⎞ ln (1 + r ) = ln ⎜ ⎟ 4 ⎝ 65 ⎠ 25
You invested in property near the River Walk in 2004 for $325,000 and sold it in 2008 for $685,000. What is the annual rate of return for this investment? 1 ⎛ 137 ⎞ ln (1 + r ) = ln ⎜ ⎟ 4 ⎝ 65 ⎠
1+ r = e4
1 ⎛ 137 ⎞ ln ⎜ ⎟ ⎝ 65 ⎠
r = e4
− 1 = 0.20490
Answer: 20.49%
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Example A model for the number of students at Palo Alto College that have heard the latest rumor might be
N = P (1 − e −.05 d ) where P is the total number of students at Palo Alto and d is the number of days that have elapsed since the rumor began.? If P = 8000 students, how many students will know the latest rumor in two days? In four days? 27
