Problem 59 - A Graphical Approach to Precalculus with Limits

Question

Solve each logarithmic equation. Express all solutions in exact form. Support your solutions by using a calculator. $$\ln e^{x}-2 \ln e=\ln e^{4}$$

Step-by-Step Solution

Answer

The solution is \( x = 6 \).

Step 1: Apply Logarithmic Identity

Recall that the natural logarithm of an exponent such that the expression is of the form \( \ln e^x \) simplifies to \( x \). Therefore, the given expression \( \ln e^{x} - 2 \ln e \) can be rewritten as \( x - 2 \ln e \) because \( \ln e \) equals 1.

Step 2: Simplify the Equation

Using the result from Step 1, the equation simplifies to: \[ x - 2 \ln e = \ln e^{4} \] Since \( \ln e = 1 \), the equation becomes: \[ x - 2 = 4 \] because \( \ln e^4 = 4 \) as well.

Step 3: Solve for x

To isolate \( x \), add 2 to both sides of the equation:\[ x - 2 + 2 = 4 + 2 \]This simplifies to:\[ x = 6 \]

Step 4: Verify the Solution

To verify, substitute \( x = 6 \) back into the original equation:\[ \ln e^{6} - 2 \ln e = \ln e^{4} \].Simplifying both sides, since \( \ln e^6 = 6 \) and \( -2 \ln e = -2 \), we have \( 6 - 2 = 4 \), which confirms that each side of the equation is equal.

Key Concepts

Natural Logarithm

The natural logarithm, abbreviated as \( \ln \), is a special logarithm with base \( e \), where \( e \) is an irrational and transcendental number approximately equal to 2.71828. This logarithm is commonly used in many areas of mathematics and science because of its natural occurrence in growth processes and wave behaviors.
Understanding \( \ln \) is crucial when dealing with exponential functions because it allows us to move between the exponential and logarithmic forms seamlessly. One of the most useful properties of the natural logarithm is that \( \ln(e^x) = x \) for any real number \( x \), and this is because raising \( e \) to the \( x \) power and then taking the logarithm with base \( e \) will simplify to \( x \) itself.

• For example, \( \ln(1) = \ln(e^0) = 0 \) since any number raised to the zero power equals 1.
• Similarly, \( \ln(e) = 1 \) because \( e^1=e \).

This property is vital for simplifying equations involving natural logarithms.

Exponents

Exponents are a way to express repeated multiplication of a number by itself. In any expression of the form \( a^b \), \( a \) is the base, and \( b \) is the exponent. Exponents follow specific rules that make calculations easier.
Some important properties of exponents that are commonly used include:

• The product of powers rule: \( a^m \cdot a^n = a^{m+n} \).
• The power of a power rule: \( (a^m)^n = a^{mn} \).
• The power of a product rule: \( (ab)^n = a^n \cdot b^n \).

Exponential functions and the number \( e \) are frequently associated because they provide a natural growth pattern, making them relevant for logarithmic equations where simplification of expressions like \( \ln(e^x) \) to \( x \) is possible. This kind of relation often simplifies the evaluation and solving of equations like the one found in the step-by-step solution.

Simplifying Equations

Simplifying equations is a fundamental step in solving algebraic problems, making the calculations more manageable and easier to interpret. When working with logarithmic and exponential equations, simplification often involves using properties of logs and exponents to reduce the expressions to a simpler form.
In the exercise given, we begin with an equation that can initially seem complex: \( \ln e^{x} - 2 \ln e = \ln e^4 \). However, by using the property \( \ln(e^x) = x \), we can simplify \( \ln e^{x} \) to \( x \).
Once you have rewritten the equation using known properties, it allows you to solve for \( x \) more easily. In this example:

• The expression \( 2\ln e \) simplifies to 2 since \( \ln e = 1 \).
• The expression \( \ln e^4 \) simplifies to 4 for the same reason.

Thus, you solve \( x - 2 = 4 \), which further simplifies to the solution \( x = 6 \). Such simplifications help maintain accuracy and improve understanding of the mathematical process involved.