Algebra 2 Practice - Solving Exponential Equations Using Natural Logs (ln)

Please subscribe!    / nickperich   *Algebra 2 Practice - Solving Exponential Equations Using Natural Logs (\(\ln\))* When solving exponential equations where the variable is in the exponent, we often use natural logarithms (\(\ln\)) to isolate the variable. --- *Example 1: Solve for \( x \)* \[ e^{2x} = 10 \] *Steps:* 1. *Take the natural logarithm (\(\ln\)) on both sides:* \[ \ln(e^{2x}) = \ln(10) \] 2. *Apply the logarithm power rule:* \[ 2x \ln(e) = \ln(10) \] 3. *Since \(\ln(e) = 1\), simplify:* \[ 2x = \ln(10) \] 4. *Solve for \( x \):* \[ x = \frac{\ln(10)}{2} \] 5. *Use a calculator to approximate:* \[ x \approx \frac{2.3026}{2} = 1.1513 \] *Final Answer:* \[ x \approx 1.15 \] This means \( e^{2(1.15)} \approx 10 \), confirming the solution. --- *Example 2: Solve for \( x \)* \[ 4e^{3x} = 25 \] *Steps:* 1. *Isolate the exponential term by dividing by 4:* \[ e^{3x} = \frac{25}{4} \] 2. *Take the natural logarithm (\(\ln\)) on both sides:* \[ \ln(e^{3x}) = \ln\left(\frac{25}{4}\right) \] 3. *Apply the logarithm power rule:* \[ 3x \ln(e) = \ln\left(\frac{25}{4}\right) \] 4. *Since \(\ln(e) = 1\), simplify:* \[ 3x = \ln\left(\frac{25}{4}\right) \] 5. *Solve for \( x \):* \[ x = \frac{\ln\left(\frac{25}{4}\right)}{3} \] 6. *Use a calculator to approximate:* \[ x \approx \frac{\ln(6.25)}{3} = \frac{1.8326}{3} \approx 0.61 \] *Final Answer:* \[ x \approx 0.61 \] This means \( 4e^{3(0.61)} \approx 25 \), confirming the solution. --- *Key Takeaways:* Use \(\ln\) when dealing with equations that have base \( e \). Apply logarithm rules, especially the power rule (\(\ln(a^b) = b\ln(a)\)). Solve for \( x \) algebraically, then use a calculator for an approximation if needed. Would you like another example? 😊 I have many informative videos for Pre-Algebra, Algebra 1, Algebra 2, Geometry, Pre-Calculus, and Calculus. Please check it out: / nickperich Nick Perich Norristown Area High School Norristown Area School District Norristown, Pa #math #algebra #algebra2 #maths #math #shorts #funny #help #onlineclasses #onlinelearning #online #study