Solve 100=(212-70)e^-0.046x+70

$$100 = ( 212 - 70 ) e ^ { - 0.046 x } + 70$$

Solve for x

$$x=\frac{500\ln(71)-500\ln(15)}{23}\approx 33.796297303$$

Steps Using Definition of Logarithm

Subtract 70 from 212 to get 142.

$$100=142e^{-0.046x}+70$$

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

$$142e^{-0.046x}+70=100$$

Subtract 70 from both sides of the equation.

$$142e^{-0.046x}=30$$

Divide both sides by 142.

$$e^{-0.046x}=\frac{15}{71}$$

Take the logarithm of both sides of the equation.

$$\log(e^{-0.046x})=\log(\frac{15}{71})$$

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

$$-0.046x\log(e)=\log(\frac{15}{71})$$

Divide both sides by \log(e).

$$-0.046x=\frac{\log(\frac{15}{71})}{\log(e)}$$

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

$$-0.046x=\log_{e}\left(\frac{15}{71}\right)$$

Divide both sides of the equation by -0.046, which is the same as multiplying both sides by the reciprocal of the fraction.

$$x=\frac{\ln(\frac{15}{71})}{-0.046}$$

Solve for x (complex solution)

$$x=-\frac{i\times 1000\pi n_{1}}{23}+\frac{500\ln(71)}{23}-\frac{500\ln(15)}{23}$$

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