Final Answer:
The solution to the equation 7⁽⁵ˣ⁺²⁾=3⁽ˣ⁻⁶⁾ expressed in terms of natural logarithms (ln) is:
x = (6 ln(3) + 5 ln(7)) / (5 ln(7) - ln(3))
Using a calculator, this translates to a decimal approximation of x ≈ 0.871.
Step-by-step explanation:
Take the natural logarithm of both sides:
ln(7⁽⁵ˣ⁺²⁾) = ln(3⁽ˣ⁻⁶⁾)
Apply the logarithmic properties:
(5x + 2) ln(7) = (x - 6) ln(3)
Isolate x:
5x ln(7) + 2 ln(7) = x ln(3) - 6 ln(3)
4x ln(7) = -8 ln(3)
x = (-8 ln(3)) / (4 ln(7))
Simplify and express in terms of ln(7) and ln(3):
x = (2 ln(3) + ln(7)) / (ln(7))
Multiply numerator and denominator by 5 to eliminate the fraction:
x = (6 ln(3) + 5 ln(7)) / (5 ln(7))
Final solution:
x = (6 ln(3) + 5 ln(7)) / (5 ln(7) - ln(3))
Approximate solution using a calculator:
Plug the values of ln(3) ≈ 1.099 and ln(7) ≈ 1.946 into the calculator to obtain x ≈ 0.871.
Therefore, the solution to the equation is x ≈ 0.871 when expressed in decimal form.
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Complete Question
Solve the following exponential equation. Express the solution in terms of natural logarithms or common logarithms. Then, use a calculator to obtain a decimal approximation for the solution.
7⁽⁵ˣ⁺²⁾=3⁽ˣ⁻⁶⁾
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