Calculus of a Single Variable, Chapter 5, 5.5, Section 5.5, Problem 27

For exponential equation:2^(3-z)=625 , we may apply the logarithm property:
log(x^y) = y * log (x) .
This helps to bring down the exponent value.
Taking "log" on both sides:
log(2^(3-z))=log(625)
(3-z)* log (2) = log(625)
Divide both sides by log (2) to isolate (3-z):
((3-z) * log (2)) /(log(2))= (log(625))/(log(2))
3-z=(log(625))/(log(2))
Subtract both sides by 3 to isolate "-z":
3-z=(log(625))/(log(2))
-3 -3
------------------------------------
-z=(log(625))/(log(2)) -3
Multiply both sides by -1 to solve +z or z:
(-1)*(-z)=(-1)* [(log(625))/(log(2)) -3]


z~~-6.288 Rounded off to three decimal places.
To check, plug-in z=-6.288 in 2^(3-z)=625 :
2^(3-(-6.288))=?625
2^(3+6.288)=?625
2^(9.288)=?625
625.1246145~~625 TRUE

Conclusion: z~~-6.288 as the final answer.

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