Problem 25 - Introductory and Intermediate Algebra for College Students 4th

Question

Solve each exponential equation by taking the logarithm on both sides. Express the solution set in terms of logarithms. Then use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. $$5 e^{x}=25$$

Step-by-Step Solution

Answer

The solution in terms of logarithms is \(x = ln(5)\). A decimal approximation for the solution is \(x \approx 1.61\).

Step 1: Take the natural logarithm of both sides

To start solving the equation \(5e^{x} = 25\), divide both sides by 5 to isolate the exponential term \(e^{x}\) on one side to get \(e^{x} = 5\). Next, take the natural logarithm on both sides, which results in \(ln(e^{x}) = ln(5)\).

Step 2: Simplify the equation

Exploit the property of logarithms which states that \(ln(a^{b})=b*ln(a)\), to simplify the left-hand side of the equation to \(x*ln(e)\), which becomes \(x\) since \(ln(e)=1\). The new equation is \(x = ln(5)\).

Step 3: Calculate a decimal approximation

Now, use a calculator to evaluate \(ln(5)\) to the nearest hundredth place. You should get \(x \approx 1.61\).

Key Concepts

Natural Logarithms

Natural logarithms, often written as \( \ln \), are logarithms with the base \( e \), where \( e \) is approximately equal to 2.71828. This special number \( e \) is known as Euler's number, and it's one of the most important constants in mathematics because it arises naturally in many growth processes and financial calculations. Taking the natural logarithm of a number is essentially asking "to what power must we raise \( e \) to obtain this number?" For instance, in our exercise, taking the natural logarithm of both sides of the equation \( e^x = 5\) turns it into the equation \( x = \ln(5) \). This step simplifies the exponential equation into a linear form, allowing us to solve for \( x \). Understanding natural logarithms is crucial for solving exponential problems with ease, especially in calculus and higher-level algebra.

Decimal Approximation

Decimal approximation is the process of converting an expression or number with a potentially infinite decimal expansion into a shorter, more manageable form. In math problems, especially those involving logarithms, we often aim to achieve approximations to a specific decimal place, generally to make the number user-friendly and simpler to interpret. In our particular exercise, after determining the solution \( x = \ln(5) \), we are asked to find a decimal approximation. To achieve that using a calculator, we enter the expression \( \ln(5) \) to find it equals approximately 1.6094379. However, for simplicity and ease of communication, we round this number to two decimal places to get \( x \approx 1.61 \).Rounding to two decimal places involves evaluating the third decimal place to determine whether to round up or down, thereby ensuring the approximation is both accurate and concise.

Logarithmic Properties

Logarithmic properties are rules that simplify complex expressions, making it possible to handle challenging problems. One key property is the **Power Rule:**

• \( \ln(a^b) = b \cdot \ln(a) \), which allows us to bring down exponents as multipliers in expressions.

In the original exercise, when \( \ln(e^x) \) is simplified, this property allows us to write it as \( x \cdot \ln(e) \). Since \( \ln(e) = 1 \), it simplifies directly to \( x \). Other important logarithmic properties include:

• **Product Rule:** \( \ln(a \cdot b) = \ln(a) + \ln(b) \)
• **Quotient Rule:** \( \ln(a/b) = \ln(a) - \ln(b) \)

These properties reduce the complexity of exponential equations and help when solving for variables, particularly in advanced algebra and calculus contexts. Understanding and effectively using these properties can turn a complex mathematical problem into one that is significantly easier to solve.