Problem 18 - Precalculus Mathematics for Calculus

Question

Express the equation in logarithmic form. (a) \(e^{x+1}=0.5\) (b) \(e^{0.5 x}=t\)

Step-by-Step Solution

Answer

(a) \(x = \ln(0.5) - 1\), (b) \(x = \frac{\ln(t)}{0.5}\).

Step 1: Identify the Exponential Equation

The given equations are exponential equations with the base as the natural number \(e\).

Step 2: Convert to Logarithmic Form (a)

For the equation \(e^{x+1} = 0.5\), apply the definition of logarithms. The formula is \(a^b = c\) can be rewritten as \(b = \log_a(c)\). Therefore, \(e^{x+1} = 0.5\) can be rewritten as \(x + 1 = \ln(0.5)\).

Step 3: Solve for x (a)

Isolate \(x\) by subtracting 1 from both sides. Hence, \(x = \ln(0.5) - 1\).

Step 4: Convert to Logarithmic Form (b)

For the equation \(e^{0.5x} = t\), again apply the definition of logarithms. Therefore, the equation is rewritten as \(0.5x = \ln(t)\).

Step 5: Solve for x (b)

Isolate \(x\) by dividing both sides by 0.5. Thus, \(x = \frac{\ln(t)}{0.5}\).

Key Concepts

Exponential Equations

Exponential equations are equations where a constant base is raised to a variable exponent. These equations often take the general form \(a^b = c\), where \(a\) is a constant, \(b\) is a variable, and \(c\) is a result. In many mathematical scenarios, especially in scientific contexts, the base \(e\), approximately equal to 2.718, is used. This is known as the natural exponential base. Exponential equations with this base are referred to as natural exponential equations. Examples include:

• \(e^{x} = 3\)
• \(e^{2x + 1} = 10\)

These equations often arise in growth processes like population growth or radioactive decay, where the change is proportional to the amount present. Solving exponential equations typically involves converting them into a logarithmic form. This transformation is a crucial step in finding the unknown variable effectively.

Natural Logarithm

A natural logarithm is a logarithm with base \(e\). It is denoted as \(\ln(x)\) rather than \(\log_e(x)\), offering a convenient shorthand. The natural logarithm is widely used due to its properties related to exponential functions and calculus, simplifying the process of differentiating or integrating exponential equations. The natural logarithm answers the question: "To what power must \(e\) be raised to yield \(x\)?" For example, \(\ln(1) = 0\) because \(e^0 = 1\), and \(\ln(e) = 1\) because \(e^1 = e\). Properties of the natural logarithm include:

• \(\ln(ab) = \ln(a) + \ln(b)\)
• \(\ln\left(\frac{a}{b}\right) = \ln(a) - \ln(b)\)
• \(\ln(a^n) = n\ln(a)\)

These rules are particularly valuable in simplifying complex logarithmic expressions.

Converting to Logarithmic Form

Converting an exponential equation to its logarithmic form allows for easier manipulation and solution of the variable exponent. The conversion is based on the principle that any exponential equation \(a^b = c\) can be rewritten as \(b = \log_a(c)\). For natural exponential equations where the base is \(e\), this becomes \(b = \ln(c)\). ### Conversion Steps: 1. **Identify the Exponential Equation:** Recognize the base \(e\) and the structure \(e^{b} = c\). 2. **Apply the Logarithmic Definition:** Rewrite the exponent in terms of its logarithm using \(b = \ln(c)\). Once in logarithmic form, solving for the variable often involves basic algebraic manipulations, making it clear and easy to isolate the variable. This technique is especially helpful in cases where direct calculation through exponents can be cumbersome or impractical. By converting to a logarithmic form, the process of solving for unknowns becomes not only simpler but also more intuitive.