91Ó°ÊÓ

The natural logarithm, often denoted as \text{ln}, is a special type of logarithm that uses the base of the mathematical constant e, where e is approximately 2.71828. The natural logarithm is particularly useful when working with exponential functions involving the base e.

In our problem, the equation contains the base e as part of the exponential function: \(e^{1-3x}=4\). To manage the exponent, we use the natural logarithm because it effectively 'cancels out' the base e. This simplifies the equation significantly.

By taking the natural logarithm of both sides, we convert the exponential equation into a linear form that is easier to handle: \[ \text{ln}(e^{1-3x})=\text{ln}(4) \]. This step sets the groundwork for using logarithm properties and isolating variables.

Logarithm properties are rules that help us manipulate logarithmic expressions. Here are some useful properties:

• \text{Product Property}: \[ \text{ln}(ab)=\text{ln}(a)+\text{ln}(b) \]
• \text{Quotient Property}: \[ \text{ln}\bigg(\frac{a}{b}\bigg)=\text{ln}(a)-\text{ln}(b) \]
• \text{Power Property}: \[ \text{ln}(a^b) = b \times \text{ln}(a) \]
• \text{Identity Property}: \[ \text{ln}(e^a) = a \]

In our exercise, we use the Identity Property. This helps us convert \[ \text{ln}(e^{1-3x}) \] into \[ 1 - 3x \]. Thus, the equation becomes \ 1-3x=\text{ln}(4) \. This simplification is key in making the equation easier to solve.

Isolating the variable is one of the primary goals when solving equations. Basically, we want to get the variable (in this case, x) alone on one side of the equation.

Here's the process for our equation after simplification: \[ 1 - 3x = \text{ln}(4) \]

Firstly, subtract 1 from both sides: \[ -3x = \text{ln}(4) - 1 \]

Next, divide both sides by -3 to isolate x: \[ x = \frac{\text{ln}(4) - 1}{-3} \]

Now, x is expressed solely in terms of constants and natural logarithms. Always remember that each operation you perform has an inverse operation that helps isolate the variable step by step. This method is powerful and essential for solving many types of equations.