Problem 95 - Precalculus : Building Concepts and Connections

Question

Solve using any method, and eliminate extraneous solutions. $$e^{\log x}=e$$

Step-by-Step Solution

Answer

The solution to \(e^{log x} = e\) is \(x=e\).

Step 1: Understand the Expression

The expression \(e^{\log x}=e\) is defined for \(x>0\) as we're taking logarithm of \(x\). The problem is to solve for \(x\).

Step 2: Simplify the Expression

In the next step, using the property of exponentials and logarithms where they cancel each other out, simplify \(e^{\log x}\) to \(x\).

Step 3: Solve for \(x\)

After simplifying the expression, the equation to solve becomes \(x=e\).

Step 4: Evaluate for Extraneous Solutions

e (approx 2.718) is greater than 0 which fits with our initial criteria for the problem (logarithm of \(x\) for \(x>0\)). Thus \(x=e\) is not an extraneous solution.

Key Concepts

Exponential and Logarithmic Properties

Understanding exponential and logarithmic properties is crucial when solving equations involving them. Exponential functions are defined by an equation of the form y = a^x, where a is a constant, and x is the exponent. The inverse of an exponential function is a logarithm. The properties of logarithms are rooted in their definition: for any positive number a and b, where a is not equal to 1, \(\log_a b = c\) if and only if a^c = b.

These properties work well together because logarithms convert multiplication into addition, division into subtraction, powers into multiplication, and roots into division. Several key properties include:

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

When working with these properties, it's important to ensure that the bases of the logarithmic expressions match and that the arguments of the logarithms are positive since the logarithm of a negative number is not defined in real numbers. The relationship between exponents and logarithms allows us to convert between exponential and logarithmic forms, which is a powerful technique in solving equations that involve these functions.

Extraneous Solutions in Logarithms

In the context of logarithmic equations, extraneous solutions can arise when we apply operations that may introduce results which are not valid in the original equation. This typically happens when we raise both sides of an equation to a power or take a root of both sides. Since the domain of logarithmic functions is limited to positive real numbers, after algebraic manipulations, we need to check that the solutions we find actually make sense in the original equation.

To identify extraneous solutions, it is important to:

• Always check the domain conditions of the original equation after solving it.
• Verify each potential solution by substituting it back into the original equation.
• Understand that not every solution generated by algebraic manipulation will be valid, particularly if they result in taking the logarithm of a negative number or zero, which is undefined in real numbers.

The concept seems straightforward but requires careful attention to detail, as overlooking domain constraints can easily lead to incorrect conclusions about the set of valid solutions.

Properties of Natural Logarithms

The natural logarithm, denoted as \(\ln x\), is a specific instance of a logarithm with base \(e\), where \(e\) is an irrational and transcendental number approximately equal to 2.71828. Its properties closely coincide with those of general logarithms but specifically apply to the base \(e\).

Some of the key properties of natural logarithms include:

• Logarithm of 1: \(\ln(1) = 0\), because \(e^0 = 1\).
• Logarithm of \(e\): \(\ln(e) = 1\), by definition of the natural logarithm.
• Inverse Properties: For any positive number x, \(\ln(e^x) = x\) and \(e^{\ln(x)} = x\). This inverse relationship is crucial when solving equations involving natural logarithms.
• Continuous Growth: Natural logarithms are often used in contexts involving continuous growth or decay because they are the inverse of the natural exponential function, which models continuous growth.

Understanding the properties of natural logarithms is especially beneficial for solving equations in calculus and other higher-level mathematics, as well as in real-world applications such as compound interest, population growth, and radioactive decay models.