Example Question - property of 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 Equation with Natural Logarithm
The equation provided in the image is: e^(4 - 7x) + 11 = 20 To solve for x, follow these steps: 1. Subtract 11 from both sides of the equation to isolate the exponential term: e^(4 - 7x) = 20 - 11 e^(4 - 7x) = 9 2. Take the natural logarithm (ln) of both sides to eliminate the base e: ln(e^(4 - 7x)) = ln(9) 3. Use the property of logarithms that says ln(a^b) = b*ln(a): (4 - 7x) * ln(e) = ln(9) Since ln(e) = 1, you can simplify this to: 4 - 7x = ln(9) 4. Isolate the term with x: -7x = ln(9) - 4 5. Divide both sides by -7 to solve for x: x = (ln(9) - 4) / -7 Using a calculator to find the numerical value of ln(9): x ≈ (2.1972 - 4) / -7 x ≈ (-1.8028) / -7 x ≈ 0.257543 So, the solution for x is approximately 0.2575 (rounded to four decimal places).
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