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

Jun 22, 2016

$y = 1.33274 \times {10}^{\frac{- 0.767704 x}{3}}$ for $0 < x < \infty$

#### Explanation:

Supposing that $\log a = {\log}_{10} a , \ln a = {\log}_{e} a$

For $0 < x < \infty$

$y = {\log}_{e} {\left(5 x\right)}^{3} / {\log}_{e} 10 - {\log}_{e} {\left(5 x\right)}^{3} + {\log}_{e} 25$

$y {\log}_{e} 10 = \left(1 - {\log}_{e} 10\right) {\log}_{e} {\left(5 x\right)}^{3} + {\log}_{e} 25 \times {\log}_{e} 10$

${\log}_{e} {\left(5 x\right)}^{3} = \frac{y {\log}_{e} 10 - {\log}_{e} 25 \times {\log}_{e} 10}{1 - {\log}_{e} 10}$

${\left(5 x\right)}^{3} = {c}_{0} {e}^{{c}_{1} y}$

where ${c}_{0} = {e}^{- \frac{{\log}_{e} 25 \times {\log}_{e} 10}{1 - {\log}_{e} 10}}$
and ${c}_{1} = {\log}_{e} \frac{10}{1 - {\log}_{e} 10}$

Finally

$x = \frac{1}{5} {c}_{0}^{\frac{1}{3}} \times {e}^{{c}_{1} / 3 y}$

or

$x = 1.33274 \times {10}^{\frac{- 0.767704 y}{3}}$

Red $y = 3 \log \left(5 x\right) - \ln \left(5 {x}^{3}\right)$
Blue $y = 1.33274 \times {10}^{\frac{- 0.767704 x}{3}}$