What is the inverse of $y=3log(5x)-ln(5x^3)$ ? ?

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Answer 1 · 2016-06-22T14:36:50

$y = 1.33274 xx10^((-0.767704 x)/3)$ for $0 < x < oo$

Supposing that $log a = log_{10}a, ln a = log_e a$

For $0 < x < oo$

$y = log_e(5x)^3/log_e 10-log_e(5x)^3+log_e 25$

$y log_e10 = (1-log_e10)log_e(5x)^3+log_e25 xxlog_e 10$

$log_e(5x)^3=(y log_e10 - log_e25 xxlog_e 10)/ (1-log_e10)$

$(5x)^3=c_0e^{c_1y}$

where $c_0 = e^(-(log_e25 xxlog_e 10)/ (1-log_e10))$
and $c_1 = log_e10/(1-log_e10)$

Finally

$x = 1/5 c_0^{1/3} xx e^{c_1/3 y}$

or

$x = 1.33274 xx10^((-0.767704 y)/3)$

Red $y=3log(5x)-ln(5x^3)$
Blue $y = 1.33274 xx10^((-0.767704 x)/3)$

enter image source here

What is the inverse of y=3log(5x)-ln(5x^3)y=3log(5x)-ln(5x^3)? ?