Questions: Solve the equation accurate to three decimal places. 3(8^(x-4)) = 38 Solve the equation accurate to three decimal places. (1 + 0.90/365)^(365t) = 2 Find the derivative of the function. y = 2^(-3x)

Solve the equation accurate to three decimal places.

Divide both sides by 3.

\(8^{x-4} = \frac{38}{3}\)

Take the natural logarithm of both sides.

\(\ln(8^{x-4}) = \ln\left(\frac{38}{3}\right)\)

Use the power rule of logarithms.

\((x-4)\ln(8) = \ln\left(\frac{38}{3}\right)\)

Solve for x.

\(x-4 = \frac{\ln\left(\frac{38}{3}\right)}{\ln(8)}\)

Calculate the value of x.

\(x = \frac{\ln\left(\frac{38}{3}\right)}{\ln(8)} + 4\)

\(\boxed{x \approx 5.221}\)

Find the derivative of the function.

Rewrite the function using exponentials.

\(y = e^{\ln(2^{-3x})}\)

Simplify the expression.

\(y = e^{-3x\ln(2)}\)

Apply the chain rule.

\(y' = e^{-3x\ln(2)} \cdot (-3\ln(2))\)

Substitute back to the original base.

\(y' = -3\ln(2) \cdot 2^{-3x}\)

\(\boxed{y' = -3\ln(2) \cdot 2^{-3x}}\)

\(\boxed{x \approx 5.221}\)
\(\boxed{y' = -3\ln(2) \cdot 2^{-3x}}\)