Solve 125*(0,858)^t+65=70

$$125 \times ( 0,858 ) ^ { t } + 65 = 70$$

Solve for t

$$t=-2\log_{0,858}\left(5\right)\approx 21.017636531$$

Steps Using Definition of Logarithm

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

$$125\times 0,858^{t}+65=70$$

Subtract 65 from both sides of the equation.

$$125\times 0,858^{t}=5$$

Divide both sides by 125.

$$0,858^{t}=\frac{1}{25}$$

Take the logarithm of both sides of the equation.

$$\log(0,858^{t})=\log(\frac{1}{25})$$

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

$$t\log(0,858)=\log(\frac{1}{25})$$

Divide both sides by \log(0,858).

$$t=\frac{\log(\frac{1}{25})}{\log(0,858)}$$

By the change-of-base formula log(a)/log(b)=log(b,a).

$$t=\log_{0,858}\left(\frac{1}{25}\right)$$