Problem 19 - 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. $$e^{x}=5.7$$

Step-by-Step Solution

Answer

The solution to the exponential equation \(e^{x} = 5.7\) is \(x = \ln(5.7)\). In decimal approximation, correct to two places, the solution will depend on the value obtained from the calculator.

Step 1: Taking the Natural Logarithm on Both Sides

The equation is \(e^{x}=5.7\). Apply the natural logarithm, ln, on both sides which gives \(\ln(e^{x})=\ln(5.7)\).

Step 2: Simplifying the Equation

Use the property of logarithms that states \(\ln(a^b) = b \ln(a)\). This simplifies \(\ln(e^{x})\) to \(x\ln(e)\). Remember that natural logarithm of e, \(\ln(e)\), equals 1, so the equation becomes \(x = \ln(5.7)\).

Step 3: Calculating the Value of x

Evaluate the natural logarithm of 5.7 using a calculator to get the approximate value of x. Make sure to round off to two decimal places.

Key Concepts

Understanding Natural Logarithms

In mathematics, the natural logarithm is a logarithm to the base of the mathematical constant e, which is approximately equal to 2.71828. It's often denoted as \(\ln\). The natural logarithm is particularly useful when solving exponential equations that involve the constant e. By taking the natural logarithm of both sides of an equation, you can effectively "bring down" exponents, simplifying the problem.

For example, given the equation \(e^{x} = 5.7\), applying the natural logarithm to both sides produces \(\ln(e^{x}) = \ln(5.7)\). This brings us to our next important property of logarithms.

Property of Logarithms: Simplifying Exponents

The property of logarithms that allows us to simplify exponents states that \(\ln(a^b) = b \ln(a)\). In our problem, \(\ln(e^{x})\) can be rewritten as \(x \ln(e)\).

Here is how it simplifies the original equation:

• When the original equation \(e^{x} = 5.7\) is translated using natural logs, it becomes \(\ln(e^{x}) = \ln(5.7)\).
• Using the exponent rule, you substitute to get \(x \cdot \ln(e) = \ln(5.7)\).
• Knowing that \(\ln(e)\) equals 1, the equation finally simplifies to \(x = \ln(5.7)\).

This immediate simplification makes it easier to solve for x.

Calculator Approximation Made Simple

Once you've simplified the equation to \(x = \ln(5.7)\), you will need a calculator to evaluate the natural logarithm. Modern calculators have a specific button for calculating \(\ln\).

Here’s the simple step to follow:

• Enter 5.7 into your calculator.
• Press the \(\ln\) button.
• Round the result to two decimal places for precision.

For this example, entering \(\ln(5.7)\) provides an approximation of 1.74 when rounded to two decimal places. This approximation provides a practical value for x, effectively solving the equation.

Successfully Simplifying Equations

Simplifying equations, especially those that involve exponents and logarithms, helps in understanding and solving complex mathematical problems more efficiently.

When dealing with exponential equations, the process often involves several steps:

• Applying logarithms to both sides to simplify exponents.
• Using properties of logarithms like \(b \ln(a)\) to "free" the variable.
• Simplifying the resulting expression to isolate the variable.

In the exercise problem, each simplification step reduces complexity, leading to a straightforward calculation using a calculator. This process underscores the importance of using logarithmic properties to streamline solving exponential equations.