91Ó°ÊÓ

A natural logarithm, denoted as \(\text{ln}\), is the logarithm to the base \(e\), where \(e\) is an irrational and transcendental number approximately equal to 2.71828. In simpler terms, the natural logarithm of a number is the power to which \(\text{e}\) must be raised to obtain that number. For example, \(\text{ln}(70)\) means the power to which we need to raise \(\text{e}\) to get 70.
Natural logarithms are often used because they simplify many mathematical operations, especially when dealing with exponential growth and decay processes. Compared to common logarithms, which use base 10, natural logarithms are more prevalent in advanced mathematics and physics.
Applying natural logarithms to both sides of an equation, such as \(2^{z}=70\), helps in transforming an exponential equation into a linear form, making it easier to solve.

The power rule of logarithms is a critical tool for solving logarithmic equations. The rule states that for any positive real numbers \(a\) and \(b\), and a real number \(c\), \(\text{ln}(a^b)=b \text{ln}(a)\). This rule allows us to move the exponent in a logarithmic expression to become a coefficient in front of the logarithm.
For example, in the equation \(2^z = 70\), applying the natural logarithm to both sides gives us \(\text{ln}(2^z) = \text{ln}(70)\). Using the power rule, this transforms into \(z \text{ln}(2) = \text{ln}(70)\), substantially simplifying the problem by removing the exponent.

An exact solution to an equation is one that is expressed in a precise mathematical form without any approximations. For exponential equations like \(2^z=70\), the exact solution involves using logarithms. After applying the natural logarithms and using the power rule, we isolate \(\text{z}\) to find \(\text{z} = \frac{\text{ln}(70)}{\text{ln}(2)}\).
This fraction represents our exact solution in terms of natural logarithms. It is exact because it provides the relationship between the variables without rounding any numbers.

Sometimes, it’s useful to find approximate solutions to an equation for practical purposes or ease of use. After finding the exact solution \(z = \frac{\text{ln}(70)}{\text{ln}(2)}\), we can use a calculator to find approximate decimal values for the logarithms.
Using a calculator, \(\text{ln}(70) \approx 4.2485\) and \(\text{ln}(2) \approx 0.6931\). Dividing these results gives us \(z \approx \frac{4.2485}{0.6931} \approx 6.1297\).
This value is rounded to four decimal places, providing a practical approximation of the solution to the problem.