Question - Solving Exponential Equations with Natural Logarithms
mathsolving exponential equationsnatural logarithmsproperties of logarithmscalculating logarithmic values
Solution:
To solve the equation $$ e^{4-7x} + 11 = 20 $$, follow these steps:1. Subtract 11 from both sides of the equation:\[ e^{4-7x} + 11 - 11 = 20 - 11 \]\[ e^{4-7x} = 9 \]2. Now take the natural logarithm (ln) of both sides to solve for $$ 4-7x $$:\[ \ln(e^{4-7x}) = \ln(9) \]Using the property of logarithms, $$ \ln(e^y) = y $$, we have:\[ 4 - 7x = \ln(9) \]3. Solve for $$ x $$:\[ -7x = \ln(9) - 4 \]\[ x = \frac{-\ln(9) + 4}{7} \]Use a calculator to compute the value of $$ \ln(9) $$, and then compute $$ x $$.
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