- Identify and evaluate exponential functions
- Analyze logarithmic functions by identifying forms, understanding their exponential relationships, and calculating different base logarithms
- Identify the hyperbolic functions, their graphs, and basic identities
Exponential Functions
The Main Idea
- Definition of Exponential Functions:
- Form: f(x)=abxf(x)=abx
- aa is a non-zero real number (initial value)
- bb is a positive real number, b≠1b≠1 (base)
- Characteristics:
- Constant base, variable exponent
- Rapid growth compared to power functions
- Domain: all real numbers
- Range: (0,∞)(0,∞) for b>1b>1, (0,∞)(0,∞) for 0<b<10<b<1
- Evaluating Exponential Functions:
- Substitute the given value for xx
- Follow order of operations
- Pay attention to parentheses and exponents
- Laws of Exponents:
- Product of Powers: bx⋅by=bx+ybx⋅by=bx+y
- Quotient of Powers: bxby=bx−ybxby=bx−y
- Power of a Power: (bx)y=bxy(bx)y=bxy
- Power of a Product: (ab)x=axbx(ab)x=axbx
- Power of a Quotient: axbx=(ab)xaxbx=(ab)x
- Applications:
- Population growth
- Compound interest
- Radioactive decay
Given the exponential function f(x)=100·3x/2f(x)=100⋅3x/2, evaluate f(4)f(4) and f(10)f(10).
Go to World Population Balance for another example of exponential population growth.
Simplify the following expression and then evaluate it for x=2x=2:
(3x2)3⋅2x−19x−1⋅4x+2(3x2)3⋅2x−19x−1⋅4x+2
Use the laws of exponents to simplify (6x−3y2)(12x−4y5)(6x−3y2)(12x−4y5).
Logarithmic Functions
The Main Idea
- Definition of Logarithmic Functions:
- Inverse of exponential functions
- logb(x)=ylogb(x)=y if and only if by=xby=x
- Domain: (0,∞)(0,∞), Range: (−∞,∞)(−∞,∞)
- Common Logarithms:
- Base 10: log10(x)log10(x), often written as log(x)log(x)
- Natural log: loge(x)loge(x), written as ln(x)ln(x)
- Properties of Logarithms:
- Product: logb(xy)=logb(x)+logb(y)logb(xy)=logb(x)+logb(y)
- Quotient: logb(xy)=logb(x)−logb(y)logb(xy)=logb(x)−logb(y)
- Power: logb(xn)=nlogb(x)logb(xn)=nlogb(x)
- Change-of-Base Formula:
- loga(x)=logb(x)logb(a)loga(x)=logb(x)logb(a)
- Useful for calculating logs with non-standard bases
- Solving Logarithmic Equations:
- Often involves converting to exponential form
- May require using logarithm properties to simplify
Solve e2x(3+e2x)=12e2x(3+e2x)=12.
Solve ln(x3)−4ln(x)=1ln(x3)−4ln(x)=1.
Use the change-of-base formula and a calculator to evaluate log46log46.
Hyperbolic Functions
The Main Idea
- Definitions of Hyperbolic Functions:
- coshx=ex+e−x2coshx=ex+e−x2
- sinhx=ex−e−x2sinhx=ex−e−x2
- tanhx=sinhxcoshx=ex−e−xex+e−xtanhx=sinhxcoshx=ex−e−xex+e−x
- cschx=1sinhxcschx=1sinhx
- sechx=1coshxsechx=1coshx
- cothx=coshxsinhxcothx=coshxsinhx
- Graphs and Behavior:
- coshxcoshx: Similar to |ex||ex|, always ≥1≥1
- sinhxsinhx: Odd function, similar to exex for large x
- tanhxtanhx: Odd function, approaches ±1±1 as x→±∞x→±∞
- Key Identities:
- cosh2x−sinh2x=1cosh2x−sinh2x=1
- 1−tanh2x=sech2x1−tanh2x=sech2x
- coth2x−1=csch2xcoth2x−1=csch2x
- sinh(x±y)=sinhxcoshy±coshxsinhysinh(x±y)=sinhxcoshy±coshxsinhy
- cosh(x±y)=coshxcoshy±sinhxsinhycosh(x±y)=coshxcoshy±sinhxsinhy
- coshx+sinhx=excoshx+sinhx=ex
- coshx−sinhx=e−xcoshx−sinhx=e−x
- Inverse Hyperbolic Functions:
- sinh−1x=ln(x+√x2+1)sinh−1x=ln(x+√x2+1)
- cosh−1x=ln(x+√x2−1)cosh−1x=ln(x+√x2−1) (x≥1x≥1)
- tanh−1x=12ln(1+x1−x)tanh−1x=12ln(1+x1−x) (|x|<1|x|<1)
Simplify and evaluate sinh(cosh−1(3))sinh(cosh−1(3)).
Simplify cosh(2lnx).
Evaluate tanh−1(12).