Exponential and Logarithmic Functions: Fresh Take

  • 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, b1b1 (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: bxby=bx+ybxby=bx+y
    • Quotient of Powers: bxby=bxybxby=bxy
    • 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)=1003x/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)32x19x14x+2(3x2)32x19x14x+2

Use the laws of exponents to simplify (6x3y2)(12x4y5)(6x3y2)(12x4y5).

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+ex2coshx=ex+ex2
    • sinhx=exex2sinhx=exex2
    • tanhx=sinhxcoshx=exexex+extanhx=sinhxcoshx=exexex+ex
    • cschx=1sinhxcschx=1sinhx
    • sechx=1coshxsechx=1coshx
    • cothx=coshxsinhxcothx=coshxsinhx
  • Graphs and Behavior:
    • coshxcoshx: Similar to |ex||ex|, always 11
    • sinhxsinhx: Odd function, similar to exex for large x
    • tanhxtanhx: Odd function, approaches ±1±1 as x±x±
  • Key Identities:
    • cosh2xsinh2x=1cosh2xsinh2x=1
    • 1tanh2x=sech2x1tanh2x=sech2x
    • coth2x1=csch2xcoth2x1=csch2x
    • sinh(x±y)=sinhxcoshy±coshxsinhysinh(x±y)=sinhxcoshy±coshxsinhy
    • cosh(x±y)=coshxcoshy±sinhxsinhycosh(x±y)=coshxcoshy±sinhxsinhy
    • coshx+sinhx=excoshx+sinhx=ex
    • coshxsinhx=excoshxsinhx=ex
  • Inverse Hyperbolic Functions:
    • sinh1x=ln(x+x2+1)sinh1x=ln(x+x2+1)
    • cosh1x=ln(x+x21)cosh1x=ln(x+x21) (x1x1)
    • tanh1x=12ln(1+x1x)tanh1x=12ln(1+x1x) (|x|<1|x|<1)

Simplify and evaluate sinh(cosh1(3))sinh(cosh1(3)).

Simplify cosh(2lnx).

Evaluate tanh1(12).