Example Question - natural logarithms
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Solving Exponential Equations with Natural Logarithms
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.
Solving Exponential Equations with Natural Logarithms
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 \( x = \frac{4 - \ln(9)}{7} \).
Solving Exponential Equations with Natural Logarithms
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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