\( 800 = 500 e^{6r} \Rightarrow \frac{8}{5} = 1.6 = e^{6r} \Rightarrow \ln(1.6) = 6r \)
["Solving the Exponential Equation: How to Solve ( 800 = 500 e^{6r} ) Step-by-Step", "Understanding how to solve exponential equations is essential in math, science, and finance. One common problem involves equations like:", "[\n800 = 500 e^{6r}\n]", "In this article, we’ll walk through the step-by-step solution of this equation using logarithms, explaining how to transform it into a solvable form and arrive at the expression ( \ln(1.6) = 6r ).", "---", "### Understanding the Equation", "Start with the given equation:", "[\n800 = 500 e^{6r}\n]", "The goal is to solve for ( r ). Since the variable appears in the exponent, we isolate the exponential term first.", "---", "### Step 1: Isolate the exponential expression", "Divide both sides by 500:", "[\n\frac{800}{500} = e^{6r}\n]", "Simplify the fraction:", "[\n1.6 = e^{6r}\n]", "---", "### Step 2: Take the natural logarithm of both sides", "To eliminate the exponential, take the natural logarithm (ln) of both sides:", "[\n\ln(1.6) = \ln(e^{6r})\n]", "Recall that ( \ln(e^{x}) = x ), so:", "[\n\ln(1.6) = 6r\n]", "---", "### Step 3: Solve for ( r )", "Now, divide both sides by 6:", "[\nr = \frac{\ln(1.6)}{6}\n]", "This is the exact solution for ( r ).", "---", "### Why This Matters", "This type of equation appears widely in compound interest calculations, population growth models, and radioactive decay. Understanding how to manipulate exponentials and apply logarithmic functions allows precise solutions in real-world scenarios.", "---", "### Final Result", "[\n800 = 500 e^{6r} \Rightarrow \frac{8}{5} = 1.6 = e^{6r} \Rightarrow \ln(1.6) = 6r \Rightarrow r = \frac{\ln(1.6)}{6}\n]", "---", "If you want to compute a decimal approximation, use a calculator:", "- ( \ln(1.6) \approx 0.47000 )\n- So, ( r \approx \frac{0.47000}{6} \approx 0.0783 )", "---", "Summary\nTo solve ( 800 = 500 e^{6r} ):\n1. Divide both sides by 500 to get ( 1.6 = e^{6r} )\n2. Take the natural log to get ( \ln(1.6) = 6r )\n3. Solve for ( r ): ( r = \frac{\ln(1.6)}{6} )", "Mastering these steps equips you to solve many exponential equations efficiently.", "---", "Keywords:\nexponential equation solution, solve (800 = 500 e^{6r}), natural logarithm applied to exponential, step-by-step math, how to solve (e^{6r} = 1.6), logarithmic steps to find (r)", "---", "Feel free to share or bookmark this guide for quick reference when tackling exponential equations!"]
