Question - Solving Exponential Equations with Natural Logarithms
Solution:
To solve the equation provided in the image, we should isolate the exponential term on one side:$$ e^{4-7x} + 11 = 20 $$Subtract 11 from both sides:$$ e^{4-7x} = 20 - 11 $$$$ e^{4-7x} = 9 $$Now take the natural logarithm (ln) of both sides to remove the base e:$$ \ln(e^{4-7x}) = \ln(9) $$By the property of logarithms, $$\ln(e^{y}) = y$$, so:$$ 4 - 7x = \ln(9) $$Now to isolate x, we first subtract 4 from both sides:$$ -7x = \ln(9) - 4 $$Finally, we divide by -7:$$ x = \frac{\ln(9) - 4}{-7} $$$$ x = \frac{-4 - \ln(9)}{7} $$This is the solution for x in terms of natural logarithms.
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