Question

Show that \(\ln \left(1+e^{2 x}\right)=2 x+\ln \left(1+e^{-2 x}\right)\)

Short answer

\(\ln(1+e^{2 x}) = 2 x+\ln(1+e^{-2 x})\) has been shown to hold true.

Step-by-step solution

Rewrite the Left Side

Start by letting us denote the left side as a function: \[\text{Let} \, f(x) = \ln(1 + e^{2x})\]

Analyze the Right Side

Rewrite the right side in a similar manner: \[g(x) = 2x + \ln(1 + e^{-2x})\]

Simplify the Right Side Expression

Notice that the term inside the logarithm can be simplified, even further if we write the exponential term with a positive exponent: \[2x + \ln(\frac{e^{2x}+ 1}{e^{2x}}) = 2x + \ln(e^{-2x} (e^{2x} + 1))\]

Apply the Logarithm Rules

Use the property of logarithms \( \ln(ab) = \ln(a) + \ln(b)\): \[2x + \ln(e^{-2x}) + \ln(1 + e^{2x}) = 2x - 2x + \ln(1 + e^{2x})\]

Combine and Simplify Terms

When combining and simplifying terms, they cancel out leading to:\[0 + \ln(1+e^{2x}) = \ln(1+e^{2x})\]

Conclusion

Since the left side equals the right side, hence they are proved to be equal: \[\ln(1+e^{2 x}) = 2 x+\ln(1+e^{-2 x})\]

Key concepts

Natural Logarithm

A natural logarithm, denoted as \(\text{ln} \), is the logarithm to the base \( e \), where \( e \) is an irrational constant approximately equal to 2.71828. This means that the natural logarithm of a number \( x \) is the power to which \( e \) must be raised to obtain \( x \). For example, \( \text{ln}(e) = 1 \) because \( e^1 = e \).
Natural logarithms are particularly useful in calculus and higher-level mathematics because they simplify the process of differentiation and integration of exponential functions.
The natural logarithm has several important properties, such as \( \text{ln}(1) = 0 \) and \( \text{ln}(ab) = \text{ln}(a) + \text{ln}(b) \). These properties make it easier to manipulate and simplify expressions involving logarithms.

Logarithm Properties

Understanding the properties of logarithms is crucial for solving logarithmic equations and simplifying expressions.
Here are some fundamental properties:

• Product Property: \( \text{ln}(xy) = \text{ln}(x) + \text{ln}(y) \)
• Quotient Property: \( \text{ln}(\frac{x}{y}) = \text{ln}(x) - \text{ln}(y) \)
• Power Property: \( \text{ln}(x^y) = y \text{ln}(x) \)

These properties are useful for breaking down complex logarithmic expressions into simpler components.
For example, in our exercise, we used the property \( \text{ln}(ab) = \text{ln}(a) + \text{ln}(b) \) to simplify \( \text{ln}(e^{-2x}(1 + e^{2x})) \).
By applying these properties correctly, we can transform and simplify logarithmic equations to make them easier to handle.

Simplifying Expressions

Simplifying expressions is a process of transforming a mathematical expression into a simpler or more convenient form without changing its value.
This is done by applying algebraic rules and properties. In our exercise, we simplified the expression \( 2x + \text{ln}(\frac{1 + e^{2x}}{e^{2x}}) \).
We used properties of exponents and logarithms: converting \( e^{-2x} \) to \( \frac{1}{e^{2x}} \), and then applying the quotient property of logarithms.
Simplifying expressions often involves combining like terms, canceling out terms, or factoring.
It helps to recognize common patterns and properties which make the expressions easier to work with.
For instance, in our example, combining \( 2x \) with \( -2x \) resulted in a simple expression \( \text{ln}(1 + e^{2x}) \).

Algebraic Manipulation

Algebraic manipulation involves the use of algebraic techniques to rearrange, simplify, or solve equations. In our provided solution, we performed several algebraic manipulations.
These included:

• Rewriting expressions
• Applying logarithm properties
• Combining like terms
• Utilizing exponential properties

The goal is to transform equations in a way that makes them easier to handle or proves a certain property, as seen in the proof exercise involving natural logarithms.
By mastering algebraic manipulation, you can solve complex equations more efficiently and understand mathematical relationships more deeply.
In our example, careful algebraic manipulation led to simplifying and proving that \( \text{ln}(1+e^{2x}) = 2x+ \text{ln}(1+ e^{-2x}) \).