Exponential and Log Functions
Created
Tags
@May 3, 2022 8:49 AM
Calculus
logay = x
of
y = ax
The Natural Logarithm
The log base e is known as the natural logarithm
loge y = x
of
y = ex
Also in the forms of
x = Iny
x = logy
Observing the Graph of loga x
Exponential and Log Functions
1
→ Domain: x > 0
→ Range: y
∈R
→ When 0 < a < 1, the loga x is flipped along the x axis
This is because log 1 x
a
= −loga x
Log Laws
*Remember that they can go both ways
⁍
⁍
⁍
⁍
⁍
⁍
⁍
And...
1
x
loga
1
= 12 loga x1 = − 12 log a x = − log a x 2 = log a
1
x
1
2
*Above is an example, it doesn’t need to be square root and it doesn’t need to be x1
Same goes for...
log a
1
x
= − log a x
as log 1 x
a
= −loga x
*For above: it needs to be x1 or 1a
Derivative Rules for Exponential Functions
In the case of f(x)
= bx
Exponential and Log Functions
2
f ′ (x) = b [
x
bh − 1
lim
h→0
]
h
lim bh −1
h
So to remove bx , we can have f ′ (0), which simplifies b0 into 1, giving us f ′(0) = h→ 0
So, this means: f ′ (x)
= bx ⋅ f ′ (0)
Euler’s Number (The Natural Exponential Function)
To determine a base b for the exponential function f ′ (0) = 1, we can use 0.0001 for h to represent it approaching
zero, and find a b value that most nearly produces 1
The nearest should be b
= 2.7, producing an estimate of 0.9933, which is the Euler’s Number
e≐2.718⩾81
lim eh − 1
h →0
And in the case of f(x)
h
=1
= ex , its derivative remains to be f ′ (x) = ex
log e x = ln x
ex D : x ∈ R
log D : x > 0
When it comes to ln, you use e
such as ln x
y
and e
= y, because ln is basically log e x
=x
Derivative of the Natural Logarithm
*x = 0
y = ln x
y′ =
1
x
Generalized Derivative of the Natural Logarithm
*x = 0
y(x) = ln [g(x)]
1
y′ =
⋅ g′ (x)
g(x)
Derivative of an Euler’s Number
y = eg(x)
y′ = eg(x) ⋅ g′ (x)
and if it’s not Euler’s number
Exponential and Log Functions
3
Derivative of a General Exponential Function
y = bx
y′ = bx (ln b)
& ————
y = bg(x)
y′ = bg(x) ⋅ ln b ⋅ g′ (x)
& ————
y = log b x
1
y′ =
x ln b
& ————
y = log b g(x)
y′ =
g′ (x)
g(x) ln b
Logarithmic Differentiation
More Logarithmic Rules
ab = eb ln(a)
Exponential and Log Functions
4