Logarithmic, Exponential and Hyperbolic Functions

Question 1

Natural Log

Definition

Answer 1

ln(x) is defined such that

ln’(x) = 1/x

Question 2

What is the integral of 1/x ?

Answer 2

ln(x) + c

Question 3

ln(ab)

Answer 3

ln(ab) = ln(a) + ln(b)

Question 4

ln(1/a)

Answer 4

ln(1/a) = -ln(a)

Question 5

ln(a/b)

Answer 5

ln(a/b) = ln(a) - ln(b)

Question 6

ln(a^n)

Answer 6

ln(a^n) = n*ln(a)

Question 7

What is the integral of g’(x)/g(x)

Answer 7

ln(g(x)) + c

Question 8

What are the domain and range of ln(x)

Answer 8

domain: (0,∞)
range: R

Question 9

Exponential

Definition

Answer 9

exp is defined as the inverse of ln(x)
exp(ln(x)) = x
ln(exp(x)) = x

Question 10

What are the domain and range of the exponential function?

Answer 10

domain: R
range: (0,∞)

Question 11

What is the differential of the exponential function?

Answer 11

exp’(x) = exp(x)

Question 12

exp(a+b)

Answer 12

exp(a+b) = exp(a)*exp(b)

Question 13

exp(nx)

Answer 13

ep(nx) = (exp(x))^n

Question 14

Formula for x^n in terms of exp and log

Answer 14

x^n = e^(nln(x)) = exp(nln(x))

Question 15

Differential of n^x

Answer 15

d/dx(n^x) = n^x * log(n)

Question 16

cosh(x)

Answer 16

cosh(x) = (e^x + e^-x)/2

Question 17

sinh(x)

Answer 17

sinh(x) = (e^x - e^-x)/2

Question 18

cosh ²(x)

Answer 18

cosh ²(x) = 1/2 + 1/2cosh(x)

Question 19

sinh ²(x)

Answer 19

sinh ²(x) = -1/2 + cosh(2x)/2

Question 20

cosh ²(x) - sinh ²(x)

Answer 20

coosh ²(x) - sinh ²(x) = 1

Question 21

cosh(2t)

Answer 21

cosh(2t) = 2cosh ²(t) - 1

Question 22

sinh(2t)

Answer 22

sinh(2t) = 2sinh ²(t) +1

Question 23

cosh(a*b)

Answer 23

cosh(a*b) = cosh(a)cosh(b) + sinh(a)sinh(b)

Question 24

sech(x)

Answer 24

1/cosh(x)

Question 25

tanh(x)

Answer 25

sinh(x) / cosh(x) = (e^x-e^-x)/(e^x+e^-x)

Question 26

coth(x)

Answer 26

cosh(x) / sinh(x)

Question 27

What is the differential of sinhx ?

Answer 27

sinh'(x) = cosh(x)

Question 28

What is the differential of cosh(x) ?

Answer 28

cosh'(x) = sinh(x)

Question 29

Inverse Hyperbolic Functions

Answer 29

- sinh and tanh are both injectve so an inverse can be defined - cosh is not an injective function so arccosh is defined over a domain [1,∞) and range [0,∞) - each inverse function can be written explicitly in terms of ln

Question 30

arcsinh(x)

Answer 30

arcsinh(x) = ln(x + √(x²+1) )

Question 31

arccosh(x)

Answer 31

arccosh(x) = ln(x + √(x²-1) )

Question 32

What is the differential of arccosh(x) ?

Answer 32

arccosh'(x) = 1/ √(x²-1)

Question 33

What is the differential of arcsinh(x)?

Answer 33

arcsinh'(x) = 1/ √(x²+1)

Question 34

What is the differential of arctanh(x)?

Answer 34

arctanh(x) = 1/(1-x²)