Solve 61=3490(0.915)^x-2-109

$$61 = 3490 ( 0.915 ) ^ { x - 2 } - 109$$

Solve for x

$$x=-\log_{0.915}\left(\frac{13960000}{569313}\right)\approx 36.017981429$$

Steps Using Definition of Logarithm

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

$$3490\times 0.915^{x-2}-109=61$$

Add 109 to both sides of the equation.

$$3490\times 0.915^{x-2}=170$$

Divide both sides by 3490.

$$0.915^{x-2}=\frac{17}{349}$$

Take the logarithm of both sides of the equation.

$$\log(0.915^{x-2})=\log(\frac{17}{349})$$

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

$$\left(x-2\right)\log(0.915)=\log(\frac{17}{349})$$

Divide both sides by \log(0.915).

$$x-2=\frac{\log(\frac{17}{349})}{\log(0.915)}$$

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

$$x-2=\log_{0.915}\left(\frac{17}{349}\right)$$

Add 2 to both sides of the equation.

$$x=\frac{\ln(\frac{17}{349})}{\ln(\frac{183}{200})}-\left(-2\right)$$

Solve for x (complex solution)

$$x=\frac{i\times 2\pi n_{1}}{\ln(0.915)}-\log_{0.915}\left(\frac{13960000}{569313}\right)$$

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