How to Solve Exponential Equations with Base (e): A Comprehensive Guide

Exponential equations involving the base (e) can often be simplified using the properties of natural logarithms. This guide will walk you through the process of solving the equation (e^x 4). We will explore multiple methods and their underlying principles to ensure a thorough understanding of the topic.

Solving (e^x 4) Using Natural Logarithms

To solve the equation (e^x 4), the natural logarithm (ln) is the appropriate tool to use, as it is the inverse function of the exponential function with base (e).

Step-by-Step Solution

Taking the natural logarithm of both sides:

ln(e^x) ln4

Applying the property of logarithms (ln(e^x) x):

x ln4

Calculating the value:

If needed, the approximate value of ln4 can be calculated using a calculator:

x ≈ 1.386

Therefore, the solution to the equation (e^x 4) is:

x ln4 ≈ 1.386

Additional Methods and Properties

Inverse Function Property

The inverse function to (e^x) is the natural logarithm (ln x). Applying this to the given equation:

Method 1:

Take the natural logarithm of both sides:

ln(e^x) ln4

Simplify using the property (ln(e^x) x):

x ln4

Calculate the value (approximation):

x ≈ 1.386

Method 2:

Take the natural logarithm of both sides:

logbase{e}{e^x} logbase{e}{4}

Simplify using the logarithm property:

x logbase{e}{4}

Approximate the value:

x ≈ 1.37 (to 3 significant figures)

Method 3:

Take the natural logarithm of both sides:

ln e^x ln 4

Apply the property of logarithms:

x ln e ln 4

Since (ln e 1), simplify:

x ln 4

Calculate the value (approximation):

x ≈ 1.38629436111989 (to 15 decimal places)

By validating:

Near the exact value of x ln4:

e^x ≈ 4 (rounded)

Conclusion

In conclusion, solving exponential equations with base (e) primarily involves the use of natural logarithms. Whether you apply the inverse function property or use the properties of logarithms, the goal is to isolate the variable (x) by transforming the exponential equation into a simpler form. The key takeaways are the use of natural logarithms and the properties of logarithms to simplify and solve these equations.