Answer:
z = (5 + ln 36) / (1 + ln 36)
Explanation:
36 e⁵⁻ᶻ − 6²ᶻ = 0
Add 6²ᶻ to both sides:
36 e⁵⁻ᶻ = 6²ᶻ
Change 36 to 6².
6² e⁵⁻ᶻ = 6²ᶻ
Divide both sides by 6².
e⁵⁻ᶻ = 6²ᶻ / 6²
Use exponent rule.
e⁵⁻ᶻ = 6²ᶻ⁻²
Take natural log of both sides.
ln e⁵⁻ᶻ = ln 6²ᶻ⁻²
Use log rule.
(5 − z) ln e = (2z − 2) ln 6
Simplify and solve.
5 − z = (2z − 2) ln 6
5 − z = (2 ln 6) z − 2 ln 6
5 + 2 ln 6 = (2 ln 6) z + z
5 + 2 ln 6 = (1 + 2 ln 6) z
z = (5 + 2 ln 6) / (1 + 2 ln 6)
You can use log rule to simplify further.
z = (5 + ln 36) / (1 + ln 36)