Problem 18 - Contemporary Precalculus

Question

Solve the equation. First express your answer in terms of natural logarithms (for instance, \(x=(2+\ln 5) /(\ln 3)) .\) Then use a calculator to find an approximation for the answer. $$e^{-9 x}=3$$

Step-by-Step Solution

Answer

Answer: The approximate solution to the equation is \(x \approx -0.122\).

Step 1: Take the natural logarithm of both sides

Taking the natural logarithm, we have: $$\ln(e^{-9x}) = \ln(3)$$

Step 2: Use the property of logarithms

We can use the property of logarithms that \(\ln(a^b) = b\ln(a)\). So, we get: $$-9x \ln(e) = \ln(3)$$

Step 3: Simplify the equation using ln(e)

Since the natural logarithm of e is 1 (\(\ln(e) = 1\)), we can simplify the equation to: $$-9x = \ln(3)$$

Step 4: Solve for x

Dividing both sides by -9, we get: $$x = -\frac{\ln(3)}{9}$$ Now, we have expressed the answer in terms of natural logarithms.

Step 5: Use a calculator to approximate the answer

Using a calculator, we find that: $$-\frac{\ln(3)}{9} \approx -0.122$$ So, the approximate solution to the equation is \(x \approx -0.122\).

Key Concepts

Solving Exponential Equations

Exponential equations are equations in which variables appear as exponents. A common strategy to solve such equations is to take the logarithm of both sides. For the equation \[ e^{-9x} = 3, \]we first apply the natural logarithm (ln) function on both sides to isolate the term with the variable.

• This transforms the original equation into a log-based equation: \[ \ln(e^{-9x}) = \ln(3). \]
• Remember, the natural logarithm is the inverse of the exponential function, making it a powerful tool to solve exponential equations.
• By taking the logarithm, we can often simplify and "linearize" equations which makes them easier to solve.

After taking the natural logarithm, the properties of logs help further simplify the problem.

Properties of Logarithms

Understanding the properties of logarithms is crucial in simplifying logarithmic equations. One important rule we used here is the power rule:

• This rule states that \( \ln(a^b) = b\ln(a) \), which allows us to pull down the exponent in the form of a coefficient.
• In the equation \[ \ln(e^{-9x}) = \ln(3), \]the exponent \(-9x\) is brought down, transforming it into: \[ -9x \ln(e) = \ln(3). \]
• Also, knowing that \( \ln(e) = 1 \) is a fundamental constant that simplifies calculations, reducing the expression to \(-9x = \ln(3)\).

These properties provide a straightforward path to solving exponential equations and are key to expressing solutions in terms of logarithms.

Using a Calculator for Approximation

After solving an equation symbolically, the next step is often to find a numerical approximation. For our equation \(-\frac{\ln(3)}{9} \), this step involves using a calculator to evaluate the natural logarithm and carry out the division.

• Start by calculating \( \ln(3) \), which gives you approximately \( 1.0986 \). This is a key intermediate value.
• Next, perform the division by \( 9 \) to get \(-\frac{1.0986}{9} \approx -0.122 \).
• Using a calculator to obtain this approximate value helps you check the reasonableness of your symbolic result and is essential for practical applications where exact values are less convenient.

Approximations are particularly useful when sharing your findings in a format that is easy to understand without requiring further mathematical interpretation.