Problem 86 - Elementary and Intermediate Algebra

Question

Solve each equation. Give the exact solution and, when appropriate, an approximation to four decimal places. $$ e^{4 x}=60 $$

Step-by-Step Solution

Answer

Exact: \( x = \frac{\ln(60)}{4} \); Approx.: \( x \approx 1.0236 \).

Step 1: Isolate the Exponential Expression

The given equation is \( e^{4x} = 60 \). The exponential expression \( e^{4x} \) is already isolated on one side of the equation.

Step 2: Apply the Natural Logarithm

To solve for \( x \), apply the natural logarithm \( \ln \) to both sides. This gives us: \[ \ln(e^{4x}) = \ln(60) \].Using the property \( \ln(e^y) = y \), the equation simplifies to:\[ 4x = \ln(60) \].

Step 3: Solve for x

Divide both sides of the equation by 4 to solve for \( x \):\[ x = \frac{\ln(60)}{4} \].

Step 4: Calculate the Exact and Approximate Values

The exact solution for \( x \) is \( \frac{\ln(60)}{4} \). To find an approximate value, use a calculator to compute \( \ln(60) \) and then divide by 4, which gives:\[ x \approx \frac{4.094344}{4} = 1.023586 \ \approx 1.0236 \text{ (rounded to four decimal places)} \].

Key Concepts

Understanding Natural Logarithms

Natural logarithms are logarithms with base \( e \), where \( e \) is an irrational constant approximately equal to 2.71828. This special number \( e \) plays a fundamental role in the calculus and analysis of exponential functions.
A natural logarithm, denoted by \( \ln(x) \), effectively "talks" about the number of times one would multiply \( e \) to get the number \( x \). Just like other logarithms, natural logarithms have useful properties:

• \( \ln(1) = 0 \) because \( e^0 = 1 \).
• \( \ln(e) = 1 \) since \( e^1 = e \).
• \( \ln(ab) = \ln(a) + \ln(b) \).
• \( \ln\left(\frac{a}{b}\right) = \ln(a) - \ln(b) \).
• \( \ln(a^b) = b \cdot \ln(a) \).

In the context of solving exponential equations, applying the natural logarithm allows us to transform the equation into a more manageable form, such as turning \( e^{4x} = 60 \) into \( 4x = \ln(60) \). This is done using the property \( \ln(e^y) = y \) for simplification.

Exploring Exponential Functions

Exponential functions, typically expressed in the form \( f(x) = a \cdot b^x \), describe situations where quantities grow or decay at consistent percentage rates. When \( a = 1 \) and \( b = e \), the function becomes \( e^x \), which is the natural exponential function.
Some characteristics of exponential functions include:

• They always pass through the point (0,1) when \( a = 1 \).
• They exhibit rapid growth or decay. For \( b > 1 \), the function grows rapidly; for \( 0 < b < 1 \), it decays rapidly.
• As \( x \) approaches infinity, \( e^x \) approaches infinity, indicating unbounded growth.
• As \( x \) approaches negative infinity, \( e^x \) approaches 0, indicating decay to zero.

In exponential equations like \( e^{4x} = 60 \), the goal is to find the unknown exponent \( x \) that makes the equation true. By leveraging properties of logarithms, solving these equations becomes more straightforward.

The Role of Mathematical Approximation

Mathematical approximation is a key tool in finding practical, usable solutions when exact results are cumbersome or impossible. It is especially relevant when dealing with irrational numbers or complex expressions.
Approximations are all about balancing precision and simplicity, striving for a value that is close enough for practical purposes. When we solved the equation \( e^{4x} = 60 \) analytically as \( x = \frac{\ln(60)}{4} \), it provided us the exact value. However, to use this value practically, such as in engineering or everyday computations, we approximate:

• Calculate \( \ln(60) \) using a calculator, yielding approximately 4.094344.
• Divide this value by 4 to get \( x \approx 1.023586 \).
• Round this off to a specified number of decimal places – in this case, four decimal places, yielding \( x \approx 1.0236 \).

Precise approximation allows us to make informed decisions and predictions with a reduced level of error.