Solve 140(1/2)^frac{t{n}}=40

$$140 ( \frac { 1 } { 2 } ) ^ { \frac { t } { n } } = 40$$

Solve for t

$$t=\left(\log_{2}\left(7\right)-1\right)n$$

$$n\neq 0$$

Steps Using Definition of Logarithm

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

$$140\times \left(\frac{1}{2}\right)^{\frac{1}{n}t}=40$$

Divide both sides by 140.

$$\left(\frac{1}{2}\right)^{\frac{1}{n}t}=\frac{2}{7}$$

Take the logarithm of both sides of the equation.

$$\log(\left(\frac{1}{2}\right)^{\frac{1}{n}t})=\log(\frac{2}{7})$$

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

$$\frac{1}{n}t\log(\frac{1}{2})=\log(\frac{2}{7})$$

Divide both sides by \log(\frac{1}{2}).

$$\frac{1}{n}t=\frac{\log(\frac{2}{7})}{\log(\frac{1}{2})}$$

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

$$\frac{1}{n}t=\log_{\frac{1}{2}}\left(\frac{2}{7}\right)$$

Divide both sides by n^{-1}.

$$t=\frac{\left(-\left(-\log_{2}\left(7\right)+1\right)\right)n}{1}$$

Solve for n

$$n=-\log_{\frac{2}{7}}\left(2\right)t$$

$$t\neq 0$$