Exponential and Logarithmic
Functions
Exponential Functions
• Vocabulary
– Exponential Function
– Logarithmic Function
– Base
– Inverse Function
– Asymptote
– Growth
– Decay
Graphing Exponential Functions
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Make a Table of Values
Enter values of X and solve for Y
Plot on Graph
Examples:
– Y = 2x
– Y = .5x
Appreciation
• Amount of function is INCREASING – Growth!
• A(t) = a * (1 + r)t
– A(t) is final amount
– a is starting amount
– r is rate of increase
– t is number of years (x)
• Example: Invest $10,000 at 8% rate – when do
you have $15,000 and how much in 5 years?
Depreciation
• Amount of function is DECREASING – Decay!
• A(t) = a * (1 - r)t
– A(t) is final amount
– a is starting amount
– r is rate of decrease
– t is number of years (x)
• Example: Buy a $20,000 car that depreciates at
12% rate – when is it worth $13,000 and how
much is it worth in 8 years?
Inverse Functions
• Reflection of function across line x = y
• Equivalent to switching x & y values
• Example:
x
0
1
2
3
y
3
• Inverse Operations
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If subtracting – add
If adding – subtract
If multiplying – divide
If dividing – multiply
6
9
12
4
5
15
18
Inverse Functions
• Steps for creating an inverse
1.
2.
3.
4.
Rewrite the equation from f(x) = to y =
Switch variables (letters) x and y
Solve equation for y (isolate y again)
Rewrite new function as f-1(x) for new y
• Example: f(x) = 2x – 3
Logarithms
• Inverse of an exponential function
• Logbx = y
– b is the base (same as exponential function)
– Transfers to: by = x
– From exponential function: bx = y
• Write logarithmic function: logby = x
• If there is no base indicated – it is base 10
• Example: log x = y
Solving & Graphing Logarithms
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Write out in exponential form: b? = x
What value needs to go in for ?
Example: log327 = ?
Graphing –
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Plot out the Exponential Function – Table of values
Switch the x and y coordinates
Domain of exponential is range of logarithm (limits)
Range of exponential is domain of logarithm (limits)
• Example: Plot 2x and then log2x
Properties of Logarithms
• Product Property: logbx + logby = logb(x*y)
– Example: log48 + log432
• Quotient Property: logbx – logby = logb(x/y)
– Example: log575 – log53
• Power Property: logbxy = y*logbx
– Example: log285
More Logarithmic Properties
• Inverse Property: logbbx = x & blogbx = x
– Example: log775
– Example: 10log 2
• Change of Base: logbx = (logax ÷ logab)
– Example: log48
– Example: log550
Solving Exponentials and Logarithms
• If the bases of two equal exponential
functions are equal – the exponents are equal
– Examples: 3x = 32
7x+2 = 72x
48x = 162
• Logarithms are the same: common logarithms
with common bases are equal
– Examples: log7(x+1) = log75
log3(2x+2) = log33x
• Logarithms with logs only on one side
– Use the properties of logarithms to solve
Logarithmic Equations (Cont)
• Examples: (Using properties of logarithms)
– Log3(x – 5) = 2
– Log2x2 = 8
log 45x – log 3 = 1
log x + log (x+9) = 1
Solving Logarithms - continued
• Exponents without common bases
– Use common log to set exponentials equal
– Use power property to bring down exponent
– Isolate the variable
– Divide out the logs – use the calculator
• Examples:
– 5x = 7
3(2x+1) = 15
6(x+1) + 3 = 12
Exponential Inequalities
• Set up equations the same but use inequality
• Solve the same as equalities
– Example: 2(n-1) > 2x106
Compounding Interest
• Interest is compounded periodically – not just
once a year
• Formula is similar to appreciation/depreciation
– Difference is in identifying the number of periods
• A(t) = a ( 1 + r/n)nt
– A(t), a and r are same as previous
– n is the number of periods in the year
Examples of Compounding
• You invest $750 at the 11% interest with different
compounding periods for 1 yr, 10 yrs and 30 yrs:
– 11% compounded annually
– 11% compounded quarterly
– 11% compounded monthly
– 11% compounded daily
Continuous Compounding
• Continuous compounding is done using e
– e is called the natural base
– Discovering e – compounding interest lab
• Equation for continuous compounding
– A(t) = a*ert
• A(t), a, r and t represent the same values as previous
• Example: $750 at 11% compounded continuously
Natural Logarithm
• Inverse of natural base, e
• Written as ln
– Shorthand way to write loge
– Properties are the same as for any other log
• Examples:
– ln e3.2
eln(x-5)
e2ln x
• Convert between e and ln
– ex = 5
ln x = 43
ln e2x+ln ex
Transforming Exponentials
Transforming Logarithms