4.1/4.2 – Exponential and Logarithmic Functions Math 140 4.1/4.2 – Exponential and Logarithmic Functions

Some things just don’t grow linearly, they grow exponentially (ex: population, compound interest).

Some things just don’t grow linearly, they grow exponentially (ex: population, compound interest). U.S. Population Source: http://en.wikipedia.org/wiki/File:US_Population,_1790_-_2011.svg

To model such behavior, we use the exponential function, 𝑓 𝑥 = 𝑏 𝑥 . 𝑏 is the base (𝑏>0, 𝑏≠1). 𝒃>𝟏 ex: 𝑓 𝑥 = 2 𝑥 𝟎<𝒃<𝟏 ex: 𝑓 𝑥 = 1 2 𝑥

Natural Exponential Base: 𝑒= lim 𝑛→∞ 1+ 1 𝑛 𝑛 ≈2 Natural Exponential Base: 𝑒= lim 𝑛→∞ 1+ 1 𝑛 𝑛 ≈2.718281828459045… (Amount of money you’d have in an account if you invested $1 at 100% interest rate per year for one year, where interest is compounded continuously.)

In general, the continuously compounded interest formula is 𝐴=𝑃 𝑒 𝑟𝑡 , and the regular compound interest formula is 𝐴=𝑃 1+ 𝑟 𝑛 𝑛𝑡 .

Properties of exponents: 𝑏 𝑚/𝑛 = 𝑛 𝑏 𝑚 = 𝑛 𝑏 𝑚 𝑏 −𝑛 = 1 𝑏 𝑛 𝑏 0 =1 𝑏 𝑥 = 𝑏 𝑦 if and only if 𝑥=𝑦 𝑏 𝑥 𝑏 𝑦 = 𝑏 𝑥+𝑦 𝑏 𝑥 𝑏 𝑦 = 𝑏 𝑥−𝑦 𝑏 𝑥 𝑦 = 𝑏 𝑥𝑦 𝑎𝑏 𝑥 = 𝑎 𝑥 𝑏 𝑥 𝑎 𝑏 𝑥 = 𝑎 𝑥 𝑏 𝑥

Ex 1. Evaluate: 81 3 4 𝑒 3 𝑒 1 2 2 3

The logarithmic function 𝑓 𝑥 = log 𝑏 𝑥 is the ______________ of the exponential function. What does that mean?

The logarithmic function 𝑓 𝑥 = log 𝑏 𝑥 is the ______________ of the exponential function. What does that mean? inverse

The logarithmic function 𝑓 𝑥 = log 𝑏 𝑥 is the ______________ of the exponential function. What does that mean? inverse

Definition 𝑦= log 𝑏 𝑥 ⟺ 𝑏 𝑦 =𝑥 (for 𝑥>0)

Graphically

Algebraically log 𝑏 𝑏 𝑥 =𝑥 𝑏 log 𝑏 𝑥 =𝑥

log 𝑥 means log 10 𝑥 ln 𝑥 means log 𝑒 𝑥

Ex 2. Evaluate. log 2 16 = log 1000 = ln 𝑒 = ln 𝑒 =

Properties of logarithms log 𝑏 𝑢 = log 𝑏 𝑣 iff 𝑢=𝑣 log 𝑏 (𝑢𝑣) = log 𝑏 𝑢 + log 𝑏 𝑣 log 𝑏 𝑢 𝑣 = log 𝑏 𝑢 − log 𝑏 𝑣 log 𝑏 𝑢 𝑟 =𝑟 log 𝑏 𝑢 log 𝑏 1 =0 log 𝑏 𝑏 =1

Ex 3. Expand: log 2 𝑥 2 𝑦 3 𝑧 5 𝑤 8

Ex 4. Expand: ln 𝑥 3 1−𝑥 𝑦 2

Solving Exponential and Logarithmic Equations Ex 5. Solve: 𝑒 −𝑥 −2𝑥 𝑒 −𝑥 =0 Ex 6. Solve: 6=4+10 𝑒 −4𝑥 Ex 7. Solve: 5 ln (𝑥+7) =15

To evaluate logs with any base, you can change them to natural logs with this formula: log 𝑏 𝑎 = ln 𝑎 ln 𝑏 ex: log 5 2 = ln 2 ln 5 ≈0.4307