Solve 47^xdiv47^21=47^45

$$47 ^ { x } \div 47 ^ { 21 } = 47 ^ { 45 }$$

Solve for x

$$x=66$$

Steps Using Definition of Logarithm

Use the rules of exponents and logarithms to solve the equation.

$$\frac{1}{130033429462229783044185156533092847}\times 47^{x}=1755511210260049172778020908173078657717675374080672665297567056535308458607$$

Multiply both sides by 130033429462229783044185156533092847.

$$47^{x}=228275143129503741495076954846893197188864748274459173176003261052016035107201719695009640374487379913987284129$$

Take the logarithm of both sides of the equation.

$$\log(47^{x})=\log(228275143129503741495076954846893197188864748274459173176003261052016035107201719695009640374487379913987284129)$$

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

$$x\log(47)=\log(228275143129503741495076954846893197188864748274459173176003261052016035107201719695009640374487379913987284129)$$

Divide both sides by \log(47).

$$x=\frac{\log(228275143129503741495076954846893197188864748274459173176003261052016035107201719695009640374487379913987284129)}{\log(47)}$$

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

$$x=\log_{47}\left(228275143129503741495076954846893197188864748274459173176003261052016035107201719695009640374487379913987284129\right)$$

Solve for x (complex solution)

$$x=\frac{2\pi n_{1}i}{\ln(47)}+66$$

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