Problem 53 - Algebra 2 and Trigonometry

Question

In \(53-56,\) find each value of \(x\) to the nearest thousandth. $$ e^{x}=35 $$

Step-by-Step Solution

Answer

The value of \(x\) is approximately 3.555.

Step 1: Understanding the Natural Exponential Equation

The equation given is an exponential equation where the base is the natural number, and the exponent is the variable. The goal is to solve for the value of the exponent, which is represented by \(x\).

Step 2: Apply the Natural Logarithm

To isolate \(x\), take the natural logarithm of both sides of the equation. This transformation uses the inverse property of the natural exponential function: \[\ln(e^x) = \ln(35)\]

Step 3: Simplify Using Logarithmic Properties

Apply the property of logarithms that states \(\ln(e^x) = x\cdot\ln(e)\), and since \(\ln(e) = 1\), we simplify the left side:\[x = \ln(35)\]

Step 4: Calculate Using a Calculator

Compute \(\ln(35)\) using a calculator to find the precise value needed. Ensure the calculator is set properly to output results to the desired decimal point. The calculation yields:\[x \approx 3.555\]

Step 5: Rounding to the Nearest Thousandth

Verify that the result is rounded to three decimal places, as instructed. The result is already in this form: \(x = 3.555\).

Key Concepts

Exponential Equations

Exponential equations are mathematical expressions where variables appear as exponents. Such equations often involve a base raised to a power that includes a variable, like in this case where we have the equation \( e^x = 35 \). The challenge lies in isolating the variable in the exponent to find its value. These equations are classified by the characteristics of their base and exponent.

• Linear Exponents: Where the variable is simply in the position of the power, such as \( e^x \).
• Non-Linear Exponents: Where the variable's coefficient, or degree, makes the equation non-linear.

To solve these equations, transforming them into a format that allows for isolation of the variable is key. This often involves taking logarithms, which helps in 'bringing down' the exponent due to their inverse nature to exponential functions.

Natural Exponential Function

The natural exponential function is represented as \( e^x \), where \( e \) is an irrational constant approximately equal to 2.71828. It holds immense significance in mathematics and real-world applications such as growth models and complex financial calculations.The function \( e^x \) is unique because:

• It is the only function that is identical to its derivative, meaning the rate of change of \( e^x \) is the same as the value of the function itself. This property is crucial in calculus as it simplifies problems involving growth and decay.
• It is continuous and smooth, always increasing for all real numbers \( x \).

When dealing with equations like \( e^x = 35 \), we need to use logarithmic principles to convert it into a solvable form. Applying the natural logarithm capitalizes on the inherent relationship \( \,\ln(e^x) = x\ln(e) = x \), thus simplifying the derivation of \( x \).

Solving Logarithmic Equations

Logarithmic equations involve terms with logarithms, requiring specific strategies for solving them. When dealing with exponential equations like \( e^x = 35 \), introducing the natural logarithm \( \ln \) is an effective approach.Applying \( \ln \) transforms the exponential equation into a linear one:

• By taking \( \ln \) of both sides of \( e^x = 35 \), it becomes \( \ln(e^x) = \ln(35) \).
• The property \( \ln(e^x) = x\ln(e) \) leads to \( x = \ln(35) \), simplifying the equation due to \( \ln(e) = 1 \).

Utilizing a calculator is crucial at this step to ensure precise computation. Ensure the calculator is providing results to the desired decimal places—in this case, to the nearest thousandth, resulting in approximately 3.555. Understanding this process is vital, as it allows for the conversion and calculation of exponential equations through the lens of logarithmic principles.