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k 3. Chapter 3 (Course Work \( 1 \leftarrow \) Solve the following exponential equation. Express the solution set in terms of natural logarithms or commor \[ 7 e^{2 \mathrm{x}}=1253 \] The solution set expressed in terms of logarithms is \( \{\square\} \). (Use a comma to separate answers as needed. Simplify your answer. Use integers or fractions for any nur

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Mar 09,2025

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The solution set is \( x = \frac{\ln(179)}{2} \).

Solución

Solve the equation \( 7e^{2x}=1253 \). Solve the equation by following steps: - step0: Solve for \(x\): \(7e^{2x}=1253\) - step1: Divide both sides: \(\frac{7e^{2x}}{7}=\frac{1253}{7}\) - step2: Divide the numbers: \(e^{2x}=179\) - step3: Take the logarithm of both sides: \(\ln{\left(e^{2x}\right)}=\ln{\left(179\right)}\) - step4: Evaluate the logarithm: \(2x=\ln{\left(179\right)}\) - step5: Divide both sides: \(\frac{2x}{2}=\frac{\ln{\left(179\right)}}{2}\) - step6: Divide the numbers: \(x=\frac{\ln{\left(179\right)}}{2}\) The solution to the exponential equation \(7e^{2x}=1253\) is \(x = \frac{\ln(179)}{2}\). Therefore, the solution set expressed in terms of natural logarithms is \(\left\{\frac{\ln(179)}{2}\right\}\).

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The Deep Dive

To isolate \( e^{2x} \), start by dividing both sides by 7: \[ e^{2x} = \frac{1253}{7} \] Next, take the natural logarithm of both sides: \[ 2x = \ln\left(\frac{1253}{7}\right) \] Now, divide by 2 to solve for \( x \): \[ x = \frac{1}{2} \ln\left(\frac{1253}{7}\right) \] So the solution set expressed in terms of logarithms is \( \left\{ \frac{1}{2} \ln\left(\frac{1253}{7}\right) \right\} \). For some fun facts: Did you know that the natural logarithm has some super cool applications in fields like biology? For example, they help model population growth, as organisms multiply exponentially. This makes it crucial for ecologists studying sustainable practices. And for those diving deeper into the world of logarithms, exploring "The Joy of x" by Steven Strogatz can be a great start. It beautifully explores mathematical concepts, including those snazzy logarithms, in a way that’s engaging and easy to understand!

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