Exponential and Logarithmic Functions

Exponential Functions

• Vocabulary– Exponential Function– Base (Common Ratio)– Growth (Appreciation)– Decay (Depreciation)– Asymptote– Inverse Function– Logarithmic Function

Graphing Exponential Functions

• 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?

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

Inverse Functions

• Reflection of function across line x = y• Equivalent to switching x & y values• Example:

• Inverse Operations– If subtracting – add– If adding – subtract– If multiplying – divide– If dividing – multiply

x 0 1 2 3 4 5

y 3 6 9 12 15 18

Inverse Functions

• Steps for creating an inverse1. Rewrite the equation from f(x) = to y =2. Switch variables (letters) x and y3. Solve equation for y (isolate y again) 4. 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

• Write out in exponential form: b? = x• What value needs to go in for ?• Example: log327 = ?• Graphing – – 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 & blogb

x = x– Example: log775

– Example: 10log 2

• Change of Base: logbx = (logax ÷ logab)– Example: log48

– Example: log550

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:

– Convert between e and ln• ex = 5 ln x = 43

– ln e3.2 eln(x-5) e2ln x ln e2x+ln ex

• Used for constant (continuous) growth or decay– Example: P 277 – Example 4 : Half life & decay

Solving Exponentials and Logarithms

• If bases of two equal exponential functions are equal – the exponents are equal– bx = by if and only if x = y

– Examples: 3x = 32 7x+2 = 72x 48x = 162

• Logarithms are the same: common logarithms with common bases are equal– logbx = logby if and only if x = y

– Examples: log7(x+1) = log75 log3(2x+2) = log33x

• Logarithms with logs only on one side– Use the properties of logarithms to solve– Example: log3(x+7) = 3

Logarithmic Equations (Cont)

• Examples: (Using properties of logarithms)– Log3(x – 5) = 2 log 45x – log 3 = 1

– Log2x2 = 8 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

Transforming Exponentials

Transforming Logarithms