Question - Solving Exponential Equations with Natural Logarithms
Solution:
To solve the equation $$ e^{4-7x} + 11 = 20 $$, let's follow these steps:1. Isolate the exponential term on one side of the equation: $$ e^{4-7x} + 11 - 11 = 20 - 11 $$ $$ e^{4-7x} = 9 $$2. To solve for $$ x $$, take the natural logarithm of both sides: $$ \ln(e^{4-7x}) = \ln(9) $$ Recall that $$ \ln(e^y) = y $$, so: $$ 4 - 7x = \ln(9) $$3. Solve for $$ x $$: $$ -7x = \ln(9) - 4 $$ $$ x = \frac{-(\ln(9) - 4)}{7} $$ $$ x = \frac{4 - \ln(9)}{7} $$Thus, the solution to the equation is $$ x = \frac{4 - \ln(9)}{7} $$.
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