Question

The following exercises are intended to derive the fundamental properties of the natural log starting from the definition ln(x)=?x1dtt, using properties of the definite integral and making no further assumptions. Pretend, for the moment, that we do not know that \(e^{x}\) is the inverse function of \(\ln (x)\), but keep in mind that \(\ln (x)\) has an inverse function defined on \((-\infty, \infty)\). Call it \(E\). Use the identity \(\ln x y=\ln x+\ln y\) to deduce that \(E(a+b)=E(a) E(b)\) for any real numbers \(a, b\).

Short answer

We proved that for any real numbers \(a\) and \(b\), \(E(a+b) = E(a)E(b)\).

Step-by-step solution

Definition Recap

The natural logarithm is defined as \( \ln(x) = \int_{1}^{x} \frac{1}{t} \, dt \). We will use this definition and properties of definite integrals to explore properties of \(\ln(x)\).

Property of Logarithms

One of the fundamental properties of logarithms is that for any positive real numbers \(x\) and \(y\), \( \ln(xy) = \ln(x) + \ln(y) \). This property is analogous to how the integral of sum can be split across intervals.

Interpret the Property

Using this logarithmic property and understanding that \( \ln(x) \) is derived from an integral, for any positive \(x\) and \(y\), the following applies:\[ \ln(x) + \ln(y) = \int_{1}^{x} \frac{1}{t} dt + \int_{1}^{y} \frac{1}{t} dt = \int_{1}^{xy} \frac{1}{t} dt = \ln(xy) \]

Define the Inverse Function

The inverse function of \( \ln(x) \) is \( E \). We do not assume \( E \) is \( e^x \), but we know \( E(E^{-1}(x)) = x \). Since \( \ln(x) \) and \( E \) are inverses, their operation results in each other's identity.

Derive the Property of Inverse Function

From the property \( \ln(xy) = \ln(x) + \ln(y) \) and given that \( E \) is the inverse of \( \ln \):\[ E(\ln(xy)) = E(\ln(x) + \ln(y)) \]Hence, the property becomes:\[ E(a+b) = E(a)E(b) \] where \( a = \ln(x) \), \( b = \ln(y) \).

Concluding the Property

Therefore, using the properties of the logarithm and integrating the inverse nature of \( E \), we conclude that for any real numbers \( a \) and \( b \), \( E(a+b) = E(a)E(b) \).

Key concepts

Definite Integrals

To truly understand natural logarithms, we must first become familiar with definite integrals. A definite integral is a mathematical concept used to calculate the area under a curve. This curve is defined by a function over a specific interval. In the case of the natural logarithm, the definite integral plays a crucial role in its definition. For natural logarithms, the function is \( rac{1}{t} \), and the interval is from 1 to x. So, the integral \( \int_{1}^{x} \frac{1}{t} \, dt \) gives us the area under the curve of \( \frac{1}{t} \) from the point 1 to x. This area under the curve provides the value of \( \ln(x) \). By understanding the area concept, it becomes easier to grasp why logarithms have certain properties. For example, the integral rule \( \int_a^b f(t) \, dt + \int_b^c f(t) \, dt = \int_a^c f(t) \, dt \) reflects the property \( \ln(xy) = \ln(x) + \ln(y) \). This property simply states that the area under the curve from 1 to xy is the sum of the areas from 1 to x and x to y. Definite integrals not only define \( \ln(x) \) but also provide insights into logarithmic properties, making them fundamental in understanding the natural logarithm.

Inverse Functions

Inverse functions may sound complex, but they're simply about reversing operations. If function \( f \) takes an input \( x \) and gives an output \( y \), then the inverse function \( f^{-1} \) takes \( y \) back to \( x \). This means \( f(f^{-1}(y)) = y \) and \( f^{-1}(f(x)) = x \). In the context of the natural logarithm, its inverse function is called \( E \). Although we're used to associating this with \( e^x \), here we consider \( E \) without assuming this relationship. We know that \( E \) undoes the action of \( \ln(x) \); therefore, when \( \ln(x) \) reaches a certain value, \( E \) can revert it back to the original number. This interaction demonstrates that these functions are inverses of each other by definition. The beauty of inverse functions lies in their symmetry. For instance, if you know \( E(a) \) and \( E(b) \) are two quantities obtained from \( \ln(x) \) and \( \ln(y) \) respectively, the property states \( E(a+b) = E(a)E(b) \). It highlights how additions and multiplications are interconnected via their inverses, linking deeply with logarithmic properties. By studying inverse functions, you gain a richer understanding of the interplay between \( \ln \) and \( E \).

Properties of Logarithms

Understanding logarithms is easier when you delve into their properties, which often mirror familiar arithmetic operations. The natural logarithm, \( \ln(x) \), is built upon definite integrals, paving the way for these essential properties. One fundamental property is \( \ln(xy) = \ln(x) + \ln(y) \). This property expresses how multiplication inside the logarithm can be translated into addition outside. This behavior is often linked to integrals since the sum of areas (definite integrals) corresponds to products being split into sums, as discussed earlier. Another significant property involves exponents: \( \ln(x^y) = y \ln(x) \). Here, raising x to the power of y transforms into multiplication, similar to how integrals manipulate scaling factors. These properties are not just number tricks; they are grounded in the behaviour of integrals and inverse functions. They provide shortcuts that help simplify complex logarithmic expressions into more manageable terms. For any aspiring math student, mastering the properties of logarithms unlocks new perspectives on exponential growth, decay, and beyond.