The logarithm functions\(: ln(x), log_a(x)\)
Both functions \(log_a(x)\) and \(ln(x)\) being defined as similar up to a coefficient \(ln(a)\), the following properties will be true for both logarithmic functions.
Definition of the natural logarithm
$$ \forall x \in \mathbb{R^+}, $$
$$ ln(x) = \int^x_1 \frac{dt}{t} $$
As well we do have:
$$ \forall x \in \hspace{0.05em} \mathbb{R}^+, $$
$$ ln(x)' = \frac{1}{x} $$
$$ $$
$$ ln(1) = 0 $$
Logarithm of a product \(: ln(ab)\)
$$ \forall (a,b) \in \hspace{0.05em} \left[ \mathbb{R}^*_+ \right]^2, $$
$$ ln(ab) = ln(a)+ ln(b) $$
Logarithm of an inverse\(: ln \left(\frac{1}{a} \right)\)
$$ \forall a \in \hspace{0.05em} \mathbb{R}^*_+, $$
$$ ln \left(\frac{1}{a} \right) = -ln(a) $$
Logarithm of a quotient\(: ln \left(\frac{a}{b} \right)\)
$$ \forall (a,b) \in \hspace{0.05em} \left[ \mathbb{R}^*_+ \right]^2, $$
$$ ln \left(\frac{a}{b} \right) = ln(a) - ln(b) $$
Logarithm of a power\(: ln \left( a^n \right)\)
$$ \forall a \in \hspace{0.05em} \mathbb{R}^*_+, \ \forall n \in \hspace{0.05em} \mathbb{R},$$
$$ ln(a^n) = n \ ln(a) $$
Link between logarithm to the base a or b and natural logarithm\( : log_a(x), log_b(x), ln(x)\)
In the general way, for two logarithms respectively to the base \(a\), \(b\) or \(e\) (natural logarithm),
$$ \forall a \in \hspace{0.05em} \Bigl[ \mathbb{R}^*_+ \hspace{0.2em} \backslash \{1\} \Bigr], \ \forall x \in \hspace{0.05em} \mathbb{R}^*_+, $$
$$ log_a(x) = \frac{ln(x) }{ln(a)} $$
$$ \forall a \in \hspace{0.05em} \Bigl[ \mathbb{R}^*_+ \hspace{0.2em} \backslash \{1\} \Bigr], \ \forall x \in \hspace{0.05em} \mathbb{R}^*_+, $$
$$ log_a(x)' = \frac{1}{x \ ln(a)} $$
$$ \forall a \in \hspace{0.05em} \Bigl[ \mathbb{R}^*_+ \hspace{0.2em} \backslash \{1\} \Bigr], \ \forall b \in \hspace{0.05em} \mathbb{R}^*_+, \ \forall x \in \hspace{0.05em} \mathbb{R}^*_+, $$
$$ log_a(x) = \frac{ log_b(x) }{log_b(a)} $$
$$ \forall (a, b) \in \hspace{0.05em} \Bigl[ \mathbb{R}^*_+ \hspace{0.2em} \backslash \{1\} \Bigr]^2, $$
$$ log_a(b) = \frac{1}{log_b(a)} $$
The exponential functions\(: e^x, a^x \)
Exponential functions, being defined as the reciprocal function of the logarithm (respectively to its base), have all the reciprocal properties of logarithms.
The definition of the exponential function
$$ \forall x \in \mathbb{R}, $$
$$ (e^x)' = e^x $$
$$ \forall x \in \mathbb{R^*_+}, $$
$$ e^{ln(x)} = x$$
$$ \forall x \in \mathbb{R}, $$
$$ ln(e^{x}) = x$$
$$ $$
$$ e^0 = 1 $$
$$ \forall (a, b) \in \hspace{0.05em} \mathbb{R}^2,$$
$$ e^a \ e^b = e^{a+b}$$
$$ \forall (a, b) \in \hspace{0.05em} \mathbb{R}^2,$$
$$ \frac{1}{e^b} = e^{- b}$$
$$ \forall (a, b) \in \hspace{0.05em} \mathbb{R}^2,$$
$$ \frac{e^a}{e^b} = e^{a - b}$$
$$ \forall (a, b) \in \hspace{0.05em} \mathbb{R}^2,$$
$$ (e^a)^b = e^{ab} $$
$$ $$
$$ e \approx 2.7182818... $$
Link between powers to the base a or b and the exponential function\(: a^x, b^x, e^x\)
$$ \forall a \in \hspace{0.05em} \mathbb{R}^*_+, \ \forall x \in \hspace{0.05em} \mathbb{R}, $$
$$ a^x = e^{x \ ln(a)} $$
$$ \forall a \in \hspace{0.05em} \mathbb{R}^*_+, \ \forall x \in \hspace{0.05em} \mathbb{R}, $$
$$ (a^x)' = a^x \ ln(a) $$
$$ \forall (a,b) \in \hspace{0.05em} \left[ \mathbb{R}^*_+ \right]^2, \ \forall x \in \hspace{0.05em} \mathbb{R}, $$
$$ a^x = b^{x \ log_b(a)} $$
$$ \forall (a,b) \in \hspace{0.05em} \left[ \mathbb{R}^*_+ \right]^2, $$
$$ a^{ln(b)} = b^{ln(a)} $$
$$ \forall (a,b, n) \in \hspace{0.05em} \left[ \mathbb{R}^*_+ \right]^3, $$
$$ a^{log_n(b)} = b^{log_n(a)}$$
Recap table of logarithm and exponential functions
Let \(a \in \mathbb{R}\) be a real number and \(f\) a function such as:
$$\forall x \in \mathbb{R}, \ f : x \longmapsto a^x$$
$$ \hspace{5.5em} \mathbb{R} \longmapsto \mathbb{R^+}$$
The origin oh the logarithm comes from the question of determining a function that returns the exponent \(x\) of a function \(f(x) = a^x\). Therefore, this function will be the reciprocal function of the function \(f\), because it returns identity \((f \circ f^{-1})\):
$$f^{-1} : a^x \longmapsto x $$
$$ \hspace{2.6em} \mathbb{R^+} \longmapsto \mathbb{R}$$
We know from the derivative of a reciprocal function that:
$$ \forall (f,f^{-1}) \in F(\mathbb{R}, \mathbb{R})^2, \enspace (f' \circ f^{-1}) \neq 0, $$
$$ ( f^{-1} )'= \frac{1}{ (f' \circ f^{-1})} $$
So,
$$ ( f^{-1} )'= \frac{1}{ ((a^x)'\circ f^{-1})} \qquad (1) $$
But, by the definition of a derivative:
$$ (a^x)' = lim_{h \to 0 } \enspace \frac{a^{x + h} - a^x}{h} $$
$$ (a^x)' = lim_{h \to 0 } \enspace \frac{a^x a^h - a^x}{h} $$
$$ (a^x)' = a^x \ lim_{h \to 0 } \enspace \frac{a^h - 1}{h} $$
If we take the hypothesis that this limit exists, and we call it \(\alpha\), we then have:
$$ \forall x \in \hspace{0.05em} \mathbb{R}^*_+, \ (a^x)' = \alpha \ a^x \qquad (2) $$
Now injecting the expression \((2)\) into \((1)\), we obtain:
$$ ( f^{-1} )'= \frac{1}{ (\alpha \ a^x \circ f^{-1})} $$
$$ ( f^{-1} )'= \frac{1}{ \alpha ( a^x \circ f^{-1})} $$
But, \(a^x = f\), so:
$$ ( f^{-1} )'= \frac{1}{ \alpha x} $$
Now, taking the antiderivative on each side of the equation:
$$ \int^x ( f^{-1}(t) )' \ dt = \int^x \frac{1}{ \alpha t} \ dt $$
Furthermore, by the reciprocity relationship:
$$ f(0) = 1 \Longleftrightarrow f^{-1}(1) = 0 $$
We thus can define this function by an integral:
$$ \int^x_1 ( f^{-1}(t) )' \ dt = \int^x_1 \frac{1}{ \alpha t} \ dt $$
$$ f^{-1}(x) - \ \underbrace{ f^{-1}(1) } _\text{\( = \ 0 \)} \ = \frac{1}{ \alpha } \int^x_1 \frac{1}{t} \ dt $$
$$ f^{-1}(x) = \frac{1}{ \alpha } \int^x_1 \frac{dt}{t} \qquad (3) $$
We no have a definition of what is the Napierian logarithm (or natural logarithm):
$$ \forall x \in \hspace{0.05em} \mathbb{R}^+, $$
$$ ln(x) = \int^x_1 \frac{dt}{t} $$
Consequently, we do also have:
$$ ln(1) = \int^1_1 \frac{dt}{t} = 0$$
$$ ln(1) = 0 $$
Moreover:
$$ \forall x \in \hspace{0.05em} \mathbb{R}^+, \ ln(x)' = \ \left( \int^x_1 \frac{dt}{t} \right)' $$
$$ ln(x)' = \frac{1}{x} $$
So,
$$ \forall x \in \hspace{0.05em} \mathbb{R}^+, $$
$$ ln(x)' = \frac{1}{x} $$
In the end, with the expression \((3)\), and for all power of \(a\):
$$ \forall x \in \hspace{0.05em} \mathbb{R}^*_+, \ log_a(x) = \frac{ln(x) }{\alpha} \qquad (4) $$
with a certain parameter \(\alpha\) still to determine.
Both \(log_a(x)\) and \(ln(x)\) functions being defined as the same thing up to a coefficient \(\alpha\), the following properties will be true for both logarithmic functions.
Using the formula found previously, we have:
$$ \forall (a,b) \in \hspace{0.05em} \left[ \mathbb{R}^*_+ \right]^2, $$
$$ ln(ab) = \int^{ab}_{1} \frac{dt}{t}$$
Now using the Chasles relation applied to integrals:
$$ ln(ab) = \int^{a}_{1} \frac{dt}{t} + \int^{ab}_{a} \frac{dt}{t}$$
By setting down a variable change for the right integral:
$$ \Biggl \{ \begin{gather*} t = a \ u \\ dt = a \ du \end{gather*} $$
$$ ln(ab) = ln(a) + \int^{b}_{1} \frac{ a \ du }{a \ u } $$
$$ ln(ab) = ln(a) + \int^{b}_{1} \frac{ du }{ u } $$
And as a result,
$$ \forall (a,b) \in \hspace{0.05em} \left[ \mathbb{R}^*_+ \right]^2, $$
$$ ln(ab) = ln(a) + ln(b) $$
$$ \forall a \in \hspace{0.05em} \mathbb{R}^*_+,$$
$$ ln \left(\frac{1}{a} \right) = \int^{\frac{1}{a}}_{1} \frac{dt}{t}$$
By setting down a variable change for this integral:
$$ \Biggl \{ \begin{gather*} t = \frac{u}{a} \\ dt = \frac{du}{a} \end{gather*} $$
$$ ln \left(\frac{1}{a} \right) = \int^{1}_{a} \frac{du}{a} \frac{a}{u}$$
$$ ln \left(\frac{1}{a} \right) = \int^{1}_{a} \frac{du}{u} $$
$$ ln \left(\frac{1}{a} \right) = -\int^{a}_{1} \frac{du}{u} $$
And finally,
$$ \forall a \in \hspace{0.05em} \mathbb{R}^*_+, $$
$$ ln \left(\frac{1}{a} \right) = -ln(a) $$
By simply taking the two previous properties, we have:
$$ \forall (a,b) \in \hspace{0.05em} \left[ \mathbb{R}^*_+ \right]^2, $$
$$ ln \left(\frac{a}{b} \right) = ln \left(a \times \frac{1}{b} \right) $$
$$ ln \left(\frac{a}{b} \right) = ln(a) - ln \left(\frac{1}{b} \right) $$
And as a result,
$$ \forall (a,b) \in \hspace{0.05em} \left[ \mathbb{R}^*_+ \right]^2, $$
$$ ln \left(\frac{a}{b} \right) = ln(a) - ln(b) $$
By retrieving the previous relation:
$$ \forall (a,b) \in \hspace{0.05em} \left[ \mathbb{R}^*_+ \right]^2,$$
$$ ln(ab) = ln(a) + ln(b) $$
And replacing \(b\) by \(a\), we obtain:
$$ \forall a \in \hspace{0.05em} \mathbb{R}^*_+, $$
$$ ln(a^2) = ln(a) + ln(a) $$
$$ ln(a^2) = 2 \ ln(a) $$
Now calculating \(ln(a^3)\):
$$ \forall a \in \hspace{0.05em} \mathbb{R}^*_+,$$
$$ ln(a^3) = ln(a^2 \times a) $$
$$ ln(a^3) = ln(a^2)+ ln(a) $$
$$ ln(a^3) = 2 \ ln(a) + ln(a) $$
$$ ln(a^3) = 3 \ ln(a) $$
We see that by a direct recurrence, we obtain:
$$ \forall a \in \hspace{0.05em} \mathbb{R}^*_+, \ \forall n \in \hspace{0.05em} \mathbb{N}, $$
$$ ln(a^n) = n \ ln(a) $$
(idem moving backwards for \( n \in \hspace{0.05em} \mathbb{Z}\))
Furthermore, if we now take \( n \in \hspace{0.05em} \mathbb{R}\), we are forced to try with another method:
$$ \forall a \in \hspace{0.05em} \mathbb{R}^*_+, \ \forall n \in \hspace{0.05em} \mathbb{R}, $$
$$ ln(a^n) = \int^{a^n}_{1} \frac{dt}{t}$$
By setting down the variable change:
$$ \Biggl \{ \begin{gather*} t = u^n \\ dt = n \times u^{n-1} \ du \end{gather*} $$
we then obtain:
$$ ln(a^n) = \int^{a}_{1} \frac{ n \times u^{n-1} }{u^n} \ du $$
$$ ln(a^n) = \int^{a}_{1} \frac{ n \times u^{n-1} }{u^{n-1} \times u} \ du $$
$$ ln(a^n) = n \int^{a}_{1} \frac{du}{u} $$
And as a result,
$$ \forall a \in \hspace{0.05em} \mathbb{R}^*_+, \ \forall n \in \hspace{0.05em} \mathbb{R},$$
$$ ln(a^n) = n \ ln(a) $$
Retrieving the previous found expression:
$$ \forall x \in \hspace{0.05em} \mathbb{R}^*_+, \ log_a(x) = \frac{ln(x) }{\alpha} \qquad (4) $$
The logarithm to the base \(a\), which is the reciprocal function of the starting function \(a^x\), is defined up to a coefficient from the natural one. Let us study the behaviour og this function for \(x = a^n\):
$$ log_a( a^n) = \frac{ln( a^n) }{\alpha} \qquad (4) $$
$$ n = n \ \frac{ln(a)}{\alpha} $$
$$ \alpha = ln(a) $$
So, having found the coefficient between both logarithms:
$$ \forall a \in \hspace{0.05em} \Bigl[ \mathbb{R}^*_+ \hspace{0.2em} \backslash \{1\} \Bigr], \ \forall x \in \hspace{0.05em} \mathbb{R}^*_+, $$
$$ log_a(x) = \frac{ln(x) }{ln(a)} $$
And in this cas, its derivative is worth:
$$ \forall a \in \hspace{0.05em} \Bigl[ \mathbb{R}^*_+ \hspace{0.2em} \backslash \{1\} \Bigr], \ \forall x \in \hspace{0.05em} \mathbb{R}^*_+, $$
$$ log_a(x)' = \left( \frac{ln(x) }{ln(a)}\right)' $$
$$ log_a(x)' = \frac{ln(x)'}{ln(a)} $$
So,
$$ \forall a \in \hspace{0.05em} \Bigl[ \mathbb{R}^*_+ \hspace{0.2em} \backslash \{1\} \Bigr], \ \forall x \in \hspace{0.05em} \mathbb{R}^*_+, $$
$$ log_a(x)' = \frac{1}{x \ ln(a)} $$
In the general case, for two logarithms respectively to the base \(a\) and \(b\), we study:
$$ \Biggl \{ \begin{gather*} log_a(a^x) = x \\ log_b(a^x) = x \ log_b(a) \end{gather*} $$
Then, combining both expressions we do quickly find that:
$$log_b(a^x) = log_a(a^x) \ log_b(a) $$
And as a result,
$$ \forall a \in \hspace{0.05em} \Bigl[ \mathbb{R}^*_+ \hspace{0.2em} \backslash \{1\} \Bigr], \ \forall b \in \hspace{0.05em} \mathbb{R}^*_+, \ \forall x \in \hspace{0.05em} \mathbb{R}^*_+, $$
$$ log_a(x) = \frac{ log_b(x) }{log_b(a)} $$
Furthermore, the previous relation and its twin lead us to:
$$ \forall (a, b) \in \hspace{0.05em} \Bigl[ \mathbb{R}^*_+ \hspace{0.2em} \backslash \{1\} \Bigr]^2, \ \forall x \in \hspace{0.05em} \mathbb{R}^*_+, $$
$$ log_a(x) = \frac{ log_b(x) }{log_b(a)} $$
$$ log_b(x) = \frac{ log_a(x) }{log_a(b)} $$
$$ \Longleftrightarrow \frac{log_a(x)}{log_b(x)} = \frac{ 1}{log_b(a)} $$
$$ \Longleftrightarrow log_a(b) = \frac{log_a(x)}{log_b(x)} $$
So,
$$ \forall (a, b) \in \hspace{0.05em} \Bigl[ \mathbb{R}^*_+ \hspace{0.2em} \backslash \{1\} \Bigr]^2, $$
$$ log_a(b) = \frac{1}{log_b(a)} $$
Now consider the reciprocal function of the function \(g(x) = ln(x)\).
If we take the formula for the derivative of a reciprocal function, we do have:
$$ \forall (g,g^{-1}) \in F(\mathbb{R}, \mathbb{R})^2, \enspace (g' \circ g^{-1}) \neq 0, $$
$$ ( g^{-1} )'= \frac{1}{ (g' \circ g^{-1})} $$
So,
$$ ( g^{-1} )'= \frac{1}{ (ln(x)' \circ g^{-1})} $$
$$ ( g^{-1} )'= \frac{1}{ \frac{1}{g^{-1}} } $$
$$ ( g^{-1} )'= g^{-1} $$
This function has itself for derivative !
It is the exponentiel function which we will note \(e^x\) but sometimes \(exp(x)\). Indeed, we will later on that it has all the properties of a power function of x.
$$ \forall x \in \mathbb{R}, $$
$$ (e^x)' = e^x $$
$$ \forall x \in \mathbb{R^*_+}, $$
$$ e^{ln(x)} = x$$
$$ \forall x \in \mathbb{R}, $$
$$ ln(e^{x}) = x$$
$$ $$
$$ e^0 = 1 $$
Having said that \(g(x) = ln(x)\) has the following property:
$$xy \longmapsto ln(xy) = ln(x) + ln(y) $$
$$ \left(\left[ \mathbb{R^*_+} \right]^2 \longmapsto \hspace{0.05em} \mathbb{R}^2\right)$$
If we set down new variables:
$$ \Biggl \{ \begin{gather*} X = ln(x) \Longleftrightarrow x = e^{X}\\ Y = ln(y) \Longleftrightarrow y = e^{Y} \end{gather*} \qquad (5) $$
Then, the reciprocal function \(g^{-1} = e^x\) must have this property:
$$ ln(xy) = X + Y \longmapsto xy = e^{X + Y} \qquad (6)$$
$$ \left(\mathbb{R}^2 \hspace{0.05em} \longmapsto \hspace{0.05em}\left[ \mathbb{R^*_+} \right]^2\right)$$
Both expressions \((5)\) and \((6)\) together show that:
$$ \forall (a, b) \in \hspace{0.05em} \mathbb{R}^2,$$
$$ e^a \ e^b = e^{a+b}$$
We know that the Taylor series of the exponential function at zero is:
$$ \forall x \in \mathbb{R}, \ e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \ ... \ + \frac{x^n}{n!}$$
Moreover, using the general formula of Taylor series at \(a\) under the form:
$$ f(a + b) = f(a) + f'(a)b + \frac{f^{(2)}(a)}{2!}b^2 \hspace{0.2em} + \hspace{0.2em} ... \hspace{0.2em} + \hspace{0.2em} \frac{f^{(n)}(a)}{n!}b^n $$
But, in the case of the exponential function, we had this property:
$$ \forall x \in \mathbb{R}, \ (e^x)' = e^x $$
Which implies that,
$$ f(a) = f'(a) = f^{(2)}(a) = \ ... \ = f^{(n)}(a) $$
So now,
$$ f(a + b) = f(a) + f(a)b + \frac{f(a)}{2!}b^2 \hspace{0.2em} + \hspace{0.2em} ... \hspace{0.2em} + \hspace{0.2em} \frac{f(a)}{n!}b^n $$
$$ f(a + b) = f(a) \underbrace{ \left(1 + b + \frac{b^2}{2!} + \frac{b^3}{3!} + \ ... \ + \frac{b^n}{n!} \right) } _\text{\( = \ f(b)\)}$$
$$ f(a + b) = f(a) f(b)$$
And as a result,
$$ \forall (a, b) \in \hspace{0.05em} \mathbb{R}^2,$$
$$ e^a \ e^b = e^{a+b}$$
Using the same logic as above,
$$ \frac{1}{y}\longmapsto ln\left(\frac{1}{y}\right) = - ln(y) $$
$$ \left(\mathbb{R^*_+} \longmapsto \hspace{0.05em} \mathbb{R}\right)$$
If we set down,
$$ Y = ln(y) \Longleftrightarrow y = e^{Y} \qquad (5) $$
So,
$$ ln \left(\frac{1}{y} \right) = - Y \longmapsto \frac{1}{y} = e^{- Y} \qquad (7)$$
$$ \left(\mathbb{R} \hspace{0.05em} \longmapsto \hspace{0.05em} \mathbb{R^*_+}\right)$$
Both expressions \((5)\) and \((7)\) together show that:
$$ \forall (a, b) \in \hspace{0.05em} \mathbb{R}^2,$$
$$ \frac{1}{e^b} = e^{- b}$$
Using the same logic as above,
$$ \frac{x}{y} \longmapsto ln\left(\frac{x}{y}\right) = ln(x) - ln(y) $$
$$ \left(\left[ \mathbb{R^*_+} \right]^2 \longmapsto \hspace{0.05em} \mathbb{R}^2\right)$$
If we set down,
$$ \Biggl \{ \begin{gather*} X = ln(x) \Longleftrightarrow x = e^{X}\\ Y = ln(y) \Longleftrightarrow y = e^{Y} \end{gather*} \qquad (5) $$
So,
$$ ln \left(\frac{x}{y} \right) = X - Y \longmapsto \frac{x}{y} = e^{X - Y} \qquad (8)$$
$$ \left(\mathbb{R}^2 \hspace{0.05em} \longmapsto \hspace{0.05em}\left[ \mathbb{R^*_+} \right]^2\right)$$
Both expressions \((5)\) and \((8)\) together show that:
$$ \forall (a, b) \in \hspace{0.05em} \mathbb{R}^2,$$
$$ \frac{e^a}{e^b} = e^{a - b}$$
Using the same logic as above,
$$ \forall n \ \mathbb{R}, \ x^n \longmapsto ln\left(x^n \right) = n \ ln(x) $$
$$ \left(\mathbb{R^*_+} \longmapsto \hspace{0.05em} \mathbb{R}\right)$$
If we set down,
$$ X = ln(x) \Longleftrightarrow x = e^{X} \qquad (5) $$
So,
$$ ln \left(x^n \right) = nX \longmapsto x^n = e^{nX} \qquad (9)$$
$$ \left(\mathbb{R} \hspace{0.05em} \longmapsto \hspace{0.05em} \mathbb{R^*_+}\right) $$
Both expressions \((5)\) and \((9)\) together show that:
$$ \forall (a, b) \in \hspace{0.05em} \mathbb{R}^2,$$
$$ (e^a)^b = e^{ab} $$
To determine an accurate value of the number \(e\), we can use the Taylor series of the exponential function at zero:
$$ \forall x \in \mathbb{R}, $$
$$ e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \ ... \ + \frac{x^n}{n!} + o(x^n)$$
$$ e^x = \sum_{k=0}^n \frac{x^k}{k!} + o(x^n)$$
Then, for \(x = 1\):
$$ e^1 = \sum_{k=0}^n \frac{1}{k!} + o(x^n) $$
This sum quickly converges towards \(2.7182818...etc\) (see table below).
$$ e \approx 2.7182818... $$
As well, we saw previously that:
$$ \forall a \in \hspace{0.05em} \Bigl[ \mathbb{R}^*_+ \hspace{0.2em} \backslash \{1\} \Bigr], \ \forall x \in \hspace{0.05em} \mathbb{R}^*_+, $$
$$ log_a(x) = \frac{ln(x) }{ln(a)} $$
Now replacing \(x\) by \(a^x\), always strictly positive on \(\mathbb{R}\):
$$ \forall a \in \hspace{0.05em} \Bigl[ \mathbb{R}^*_+ \hspace{0.2em} \backslash \{1\} \Bigr], \ \forall x \in \hspace{0.05em} \mathbb{R}, $$
$$ log_a(a^x) = \frac{ln(a^x) }{ln(a)} $$
$$ x \ ln(a) = ln(a^x) $$
Finally, applying the exponential function on each side,
$$ \forall a \in \hspace{0.05em} \Bigl[ \mathbb{R}^*_+ \hspace{0.2em} \backslash \{1\} \Bigr], \ \forall x \in \hspace{0.05em} \mathbb{R}, $$
$$ a^x = e^{x \ ln(a)} $$
With this result, we can deduce of it that the derivative of \(a^x\). Indeed, in the chapter upon the determination of the origins of the logarithmic function, we had found that:
$$ \forall x \in \hspace{0.05em} \mathbb{R}^*_+, \ (a^x)' = \alpha \ a^x \qquad (2) $$
And in the aftermath, establishing a link between the logarithm of a base a and the natural logarithm we determined that: \(\alpha = ln(a)\).
So,
$$ \forall a \in \hspace{0.05em} \mathbb{R}^*_+, \ \forall x \in \hspace{0.05em} \mathbb{R}, $$
$$ (a^x)' = a^x \ ln(a) $$
In the general case, to find the relationship between two powers of x, we retrieve the previous formula:
$$ \forall a \in \hspace{0.05em} \Bigl[ \mathbb{R}^*_+ \hspace{0.2em} \backslash \{1\} \Bigr], \ \forall b \in \hspace{0.05em} \mathbb{R}^*_+, \ \forall x \in \hspace{0.05em} \mathbb{R}^*_+, $$
$$ log_a(x) = \frac{ log_b(x) }{log_b(a)} $$
Now replacing \(x\) by \(a^x\), always strictly positive on \(\mathbb{R}\):
$$ \forall a \in \hspace{0.05em} \Bigl[ \mathbb{R}^*_+ \hspace{0.2em} \backslash \{1\} \Bigr], \ \forall b \in \hspace{0.05em} \mathbb{R}^*_+, \ \forall x \in \hspace{0.05em} \mathbb{R}, $$
$$ log_a(a^x) = \frac{ log_b(a^x) }{log_b(a)} $$
$$ log_b(a^x) = x \ log_b(a) $$
Finally, applying \(b^x\) on each side:
$$ b^{log_b(a^x)} = b^{x \ log_b(a)} $$
$$ a^x = b^{x \ log_b(a)} $$
And finally,
$$ \forall (a,b) \in \hspace{0.05em} \left[ \mathbb{R}^*_+ \right]^2, \ \forall x \in \hspace{0.05em} \mathbb{R}, $$
$$ a^x = b^{x \ log_b(a)} $$
Finally, we can also notice that:
$$ \forall (a,b) \in \hspace{0.05em} \left[ \mathbb{R}^*_+ \right]^2, $$
$$ \Biggl \{ \begin{gather*} a^{ln(b)} = \left(e^{ln(a)} \right)^{ln(b)} = e^{ln(a)ln(b)} \\ b^{ln(a)} = \left(e^{ln(b)} \right)^{ln(a)} = e^{ln(a)ln(b)} \end{gather*}$$
So finally,
$$ \forall (a,b) \in \hspace{0.05em} \left[ \mathbb{R}^*_+ \right]^2, $$
$$ a^{ln(b)} = b^{ln(a)} $$
But also for any logarithm:
$$ \forall (a,b, n) \in \hspace{0.05em} \left[ \mathbb{R}^*_+ \right]^3, $$
$$ \Biggl \{ \begin{gather*} a^{log_n(b)} \\ b^{log_n(a)} = \left(n^{log_n(b)} \right)^{log_n(a)} = n^{log_n(b)log_n(a)} = a^{log_n(b)} \end{gather*}$$
$$ a^{log_n(b)} = b^{log_n(a)}$$