We can use the Chain Rule to derive exponential functions, natural logarithms and logarithms. In this lesson we will also start applying it to finding slopes in graphs of equations.
Understanding exactly what logarithms are will help understand how we differentiate them. If its been a while since you have dealt with logarithms, review below,
The natural logarithms of a function, f(x), is symbolized by,
ln(f(x)) / or in calligraphy form 𝓵𝓷(f(x))
and means the inverse to the exponential number, e. The natural logarithm is often just referred to as the logarithm.
The different representations of logarithms and what they mean can be confusing so we will go over some of the basic ideas behind it,
The natural logarithm of e, [ln(e) or loge(e)], is 1, as e1 = e. ln(e) is like saying; e to the power of what equals e,
The natural logarithm of 1, [ln(1) or loge(1)] is 0, since e0 = 1. ln(1) is like saying, e to the power of what equals 1.
In words, think of it like:
ln(this) = what,
e to the power of this equals what
ln (ex) = x
e(ln(x)) = x
In the same way that 3x ÷ 3 = x and (x ÷ 3)(3) = x - multiplying by 3 and dividing by 3 are inverse operations to each other, in the same way that the natural logarithm and the exponential, e, are inverse operations to each other
So, if you were trying to solve an algebraic equation like,
12 = ex
Then apply the inverse to both sides of the equation, which would be the natural logarithm
ln (12) = ln(ex)
ln(12) = x
Using a calculator,
x = 2.484....
And for, Ordinary Logarithms, they are symbolized by loga(x) where loga is the inverse of ax.
So if, we had,
2x = 8,
Then, apply the inverse operation to both sides of the equation,
log2(2x) = log2(8)
x = log2(8) = 3,
Which makes sense, as 23 = 8!
Differentiate the following Natural Logarithms:
First, look at our formula for derivatives of natural logarithms with a function involved.
So for (ln(2x)) first identify what the function is in the natural logarithm.
In this case, f(x) = 2x. We will also need f'(x) = 2
Now sub these into our formula.
b) (ln(3x2 + 1))
First recall our formula for derivatives of natural logarithms
Identify our function in (ln(3x2 + 1))
In this case, f(x) = 3x2 + 1 which, when differentiated gives f'(x) = 6x
Now sub these into our formula.
View the solutions to the questions below to see examples of how to find the derivative of natural logarithmic functions.
A method we will use in our work on differentiating natural logarithms and logarithms will be the Change of Base Formula. We will be able to mathematically rewrite normal logarithms as natural logarithms, then be able to differentiate them.
In other words, for any logarithm of base a of x, we can rewrite it to equal logarithm of any positive base b of x divided by logarithm of any positive base b of a. So, if we pick that any positive base to be the exponential, e, we get,
[as loge(x) = ln (x)]
View the examples below that show how to change logarithms using the Change of Base Formula (C.B.F.)
Follow the steps in the worked examples below to how to differentiate normal logarithms
Definition - Derivative of ax for any a ≥ 0
(ax) = (ax)(ln(a)) for any a > 0
Follow the steps in the worked examples that show how to calculate derivatives
Follow the steps in the worked example below.
See below how to calculate derivatives with the aid of the Chain Rule