Solve 3140(1-0.004)+2710left(1-0.004right)^18=10040left(1-0.004right)^x

$$3140(1-0.004)+2710 { \left(1-0.004 \right) }^{ 18 } =10040 { \left(1-0.004 \right) }^{ x }$$

Solve for x

$$x=\log_{0.996}\left(\frac{8220107196093471984770874040666392094215218271}{14610122889280319213867187500000000000000000000}\right)\approx 143.495089004$$

Steps Using Definition of Logarithm

Subtract 0.004 from 1 to get 0.996.

$$3140\times 0.996+2710\left(1-0.004\right)^{18}=10040\left(1-0.004\right)^{x}$$

Multiply 3140 and 0.996 to get 3127.44.

$$3127.44+2710\left(1-0.004\right)^{18}=10040\left(1-0.004\right)^{x}$$

Subtract 0.004 from 1 to get 0.996.

$$3127.44+2710\times 0.996^{18}=10040\left(1-0.004\right)^{x}$$

Calculate 0.996 to the power of 18 and get 0.930396550661887594608597139955573614226852443362164736.

$$3127.44+2710\times 0.930396550661887594608597139955573614226852443362164736=10040\left(1-0.004\right)^{x}$$

Multiply 2710 and 0.930396550661887594608597139955573614226852443362164736 to get 2521.37465229371538138929824927960449455477012151146643456.

$$3127.44+2521.37465229371538138929824927960449455477012151146643456=10040\left(1-0.004\right)^{x}$$

Add 3127.44 and 2521.37465229371538138929824927960449455477012151146643456 to get 5648.81465229371538138929824927960449455477012151146643456.

$$5648.81465229371538138929824927960449455477012151146643456=10040\left(1-0.004\right)^{x}$$

Subtract 0.004 from 1 to get 0.996.

$$5648.81465229371538138929824927960449455477012151146643456=10040\times 0.996^{x}$$

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

$$10040\times 0.996^{x}=5648.81465229371538138929824927960449455477012151146643456$$

Divide both sides by 10040.

$$0.996^{x}=\frac{8220107196093471984770874040666392094215218271}{14610122889280319213867187500000000000000000000}$$

Take the logarithm of both sides of the equation.

$$\log(0.996^{x})=\log(\frac{8220107196093471984770874040666392094215218271}{14610122889280319213867187500000000000000000000})$$

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

$$x\log(0.996)=\log(\frac{8220107196093471984770874040666392094215218271}{14610122889280319213867187500000000000000000000})$$

Divide both sides by \log(0.996).

$$x=\frac{\log(\frac{8220107196093471984770874040666392094215218271}{14610122889280319213867187500000000000000000000})}{\log(0.996)}$$

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

$$x=\log_{0.996}\left(\frac{8220107196093471984770874040666392094215218271}{14610122889280319213867187500000000000000000000}\right)$$

Solve for x (complex solution)

$$x=\frac{i\times 2\pi n_{1}}{\ln(0.996)}+\log_{0.996}\left(\frac{8220107196093471984770874040666392094215218271}{14610122889280319213867187500000000000000000000}\right)$$

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