Solve (1/4)^18*(1/5)^m=1/2*10^35

$$( \frac { 1 } { 4 } ) ^ { 18 } \times ( \frac { 1 } { 5 } ) ^ { m } = \frac { 1 } { 2 \times 10 ^ { 35 } }$$

Solve for m

$$m=35$$

Steps Using Definition of Logarithm

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

$$\frac{1}{68719476736}\times \left(\frac{1}{5}\right)^{m}=\frac{1}{200000000000000000000000000000000000}$$

Multiply both sides by 68719476736.

$$\left(\frac{1}{5}\right)^{m}=\frac{1}{2910383045673370361328125}$$

Take the logarithm of both sides of the equation.

$$\log(\left(\frac{1}{5}\right)^{m})=\log(\frac{1}{2910383045673370361328125})$$

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

$$m\log(\frac{1}{5})=\log(\frac{1}{2910383045673370361328125})$$

Divide both sides by \log(\frac{1}{5}).

$$m=\frac{\log(\frac{1}{2910383045673370361328125})}{\log(\frac{1}{5})}$$

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

$$m=\log_{\frac{1}{5}}\left(\frac{1}{2910383045673370361328125}\right)$$