1.6 Exponential and Logarithmic Functions

Exponential and logarithmic functions are inverses of each other. Exponential functions require bases which are raised to some power. These functions are always positive, but not equal to one to avoid producing a constant function. Exponential functions increase rapidly. This is proved by the most important exponent law, which states that when two exponential functions with the same base are multiplied together the exponents are added. The number ‘e’ is approximately equal to 2.7 and is defined as being a “natural” base. Logarithmic functions also are defined with a base, but instead of being raised to a certain power, the log of that base, b is taken. The opposite of a log base ‘e’ is the natural log, ln. Hyperbolic functions create hyperbolas when graphed. Hyperbolic functions are related to trigonometric functions in that they are parities of each other. Hyperbolic functions are written just like trig functions, but with an ‘h’ added to the end. For example: cosh x.

Problems:

1. Rewrite as a whole number without using a calculator.

a) 7^0 =1

b) 10^2(2^-2 + 5^-2)=29

c) (4^3)^5/(4^5)^3 =1

d) 27^(4/3)=81

e) 8^(-1/3)*8^(5/3)=16

f) 3*4^(1/4)-12*2^(-3/2)=0

To solve this problem, I used the Laws of Exponents.

x1 = x

x0 = 1

x-1 = 1/x



xmxn = xm+n

xm/xn = xm-n

(xm)n = xmn

(xy)n = xnyn

(x/y)n = xn/yn

x-n = 1/xn



#40. Prove the addition formula for cosh x.

Cosh(x+y) = cosh x*cosh y + sinh x* sinh y

I know that cosh x is equal to ½(e^x + e^-x) and sinh x is equal to ½(e^x -e^-x). Using these values for cosh and sinh I plugged them into the addition formula and solved.