91Ó°ÊÓ

The natural logarithm, denoted as \text{ln}, is a special type of logarithm with the base \(e\). The constant \(e\) is an irrational number approximately equal to 2.71828. Natural logarithms are used frequently in mathematics, especially in calculus and solving exponential growth or decay problems. In our problem, \text{ln}(4x - 2) = 3, the \text{ln} signifies that we are dealing with a natural logarithm.
To solve the equation, we need to understand that \text{ln}(y) = x translates to \(e^x = y\). In simpler terms, if the natural logarithm of a number is given, we can rewrite it in its exponential form to make calculations easier.

An exponential equation is a type of equation in which a variable appears in the exponent. They are commonly written in the form: \(a^x = b\).
In our problem, once we isolated the logarithmic function, we rewrote \(\text{ln}(4x - 2) = 3\) as \(e^3 = 4x - 2\). This conversion is crucial because solving exponential equations often involves using natural logarithms to isolate the variable.
By transforming the original logarithmic equation into an exponential form, we can solve for \(x\). In our case, adding 2 to both sides gives us \(e^3 + 2 = 4x\). Finally, dividing both sides by 4 isolates \(x\).

In mathematics, approximations are used to simplify complex computations that cannot be solved exactly. Approximating to three decimal places means we round our answer to the nearest thousandth.
In the given problem, after solving the exponential equation, we need to approximate the value of \(e^3\). Using a calculator, we find that \(e^3 \text{ is approximately } 20.086\). Therefore, by substituting this value back in, we get \(x = \frac{20.086 + 2}{4}\). Simplifying this, we find \(x \text{ is approximately } 5.522\).
Approximations are practical as they allow us to work with simpler numerical values while still maintaining a high level of accuracy, especially useful when an exact match isn't necessary or possible.