Calculus of a Single Variable, Chapter 5, 5.5, Section 5.5, Problem 25

Problem: 3^(2x)=75 is an exponential equation.
To simplify, we need to apply logarithm property: log(x^y) = y*log(x)
to bring down the exponent that is in terms of x.
Taking "log" on both sides:
log(3^(2x))=log(75 )
(2x)log(3)=log(75)
Divide both sides by log(3) to isolate "2x ":
(2x * log (3)) /(log(3))= (log(75))/(log(3))
2x=(log(75))/(log(3))
Multiply both sides by 1/2 to isolate x:
(1/2)*2x=(log(75))/(log(3))*(1/2)
Note: You will get the same result when you divide both sides by 2.
The equation becomes:
x=(log(75))/(2log(3))
x~~1.965 Rounded off to three decimal places

To check, plug-in x=1.965 in 3^(2x)=75 :
3^(2*1.965)=?75
3^(3.93)=?75
75.0043637 ~~75 TRUE

Conclusion: x~~1.965 is the final answer.

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