Solve 127=375e^-x*30

$$127=375 { e }^{ -x \times 30 }$$

Solve for x

$$x=\frac{\ln(\frac{375}{127})}{30}\approx 0.036091298$$

Steps Using Definition of Logarithm

Divide both sides by 375.

$$\frac{127}{375}=e^{\left(-x\right)\times 30}$$

Swap sides so that all variable terms are on the left hand side.

$$e^{\left(-x\right)\times 30}=\frac{127}{375}$$

Multiply -1 and 30 to get -30.

$$e^{-30x}=\frac{127}{375}$$

Take the logarithm of both sides of the equation.

$$\log(e^{-30x})=\log(\frac{127}{375})$$

The logarithm of a number raised to a power is the power times the logarithm of the number.

$$-30x\log(e)=\log(\frac{127}{375})$$

Divide both sides by \log(e).

$$-30x=\frac{\log(\frac{127}{375})}{\log(e)}$$

By the change-of-base formula \frac{\log(a)}{\log(b)}=\log_{b}\left(a\right).

$$-30x=\log_{e}\left(\frac{127}{375}\right)$$

Divide both sides by -30.

$$x=\frac{\ln(\frac{127}{375})}{-30}$$

Solve for x (complex solution)

$$x=-\frac{\pi n_{1}i}{15}+\frac{\ln(\frac{375}{127})}{30}$$

$$n_{1}\in \mathrm{Z}$$