Problem 54 - Algebra 2 and Trigonometry

Question

In \(53-56,\) find each value of \(x\) to the nearest thousandth. $$ e^{x}=217 $$

Step-by-Step Solution

Answer

\(x \approx 5.379\)

Step 1: Understanding the Problem

We need to find the value of \(x\) such that \(e^x = 217\). To do this, we utilize logarithms, specifically natural logarithms, to solve for \(x\).

Step 2: Taking the Natural Logarithm

To solve for \(x\), we take the natural logarithm of both sides of the equation. So, we have \(x = \ln(217)\).

Step 3: Calculating the Natural Logarithm

Now, calculate \(\ln(217)\) using a calculator. This gives us the value of \(x\).

Step 4: Approximating the Value

After calculating the natural logarithm, we round the result to the nearest thousandth, which means we look at the fourth decimal place for rounding purposes.

Key Concepts

Natural Logarithms

When we talk about natural logarithms, we are dealing with a special kind of logarithm that uses the mathematical constant "e" as its base. "e" is roughly equal to 2.71828, and the natural logarithm of a number is the power to which "e" must be raised to obtain that number. For example, if we have the equation \(e^x = 217\), finding the natural logarithm of 217 actually tells us what power "e" must be raised to

• Natural logarithms are denoted as \(\ln\).
• The equation \(e^x = 217\) becomes \(x = \ln(217)\).
• Natural logarithms are particularly useful in solving equations involving exponential growth and decay, which often appear in fields like science and finance.

Understanding natural logarithms is crucial for solving exponential equations like the one given, as they allow us to "bring down" the exponent to solve for \(x\).

Solving for x

Solving for \(x\) in exponential equations typically involves isolating \(x\) on one side of the equation. Exponential equations can often seem challenging because \(x\) is an exponent, which makes typical algebraic methods ineffective. However, logarithms come to the rescue here by converting the exponential equation into a form that can be more easily solved.

• You start with an exponential equation like \(e^x = 217\).
• Apply the natural logarithm to both sides of the equation, making it \(x = \ln(217)\).
• Using a calculator, compute \(\ln(217)\) to solve for \(x\).

Once you use the natural logarithm, \(x\) is no longer in the exponent and can be computed directly, providing the solution to the equation in a manageable form.

Logarithmic Functions

Logarithmic functions serve as the inverse of exponential functions. This inversion is key because it lets us change the focus from exponentially growing functions like \(e^x\) to manageable linear calculations. By using logarithmic functions, we can solve equations involving exponential expressions.With logarithmic functions:

• They help transform an equation such as \(e^x = 217\) into a simpler equation using natural logs (\(x = \ln(217)\)).
• Logarithmic scales are used in real-world sciences like pH measurement or the Richter scale, where exponential changes in magnitude occur.
• Understanding how to use these functions ensures you can effectively solve complex equations.

Logarithmic functions convert multiplicative relationships into additive ones, allowing for simpler calculation processes, which is invaluable when working with exponential equations.