Question

Solve the following exponential equation. Express irrational solutions in exact form and as a decimal rounded to three decimal places.What is the exact answer? Select the correct choice below and, if necessary, fill in the answer box to complete your choice.What is the answer rounded to three decimal places? Select the correct choice below and, if necessary, fill in the answer box to complete your choice.

Answer 1

Solving

$2^{2x}+2^{x+1}-63=0$

Apply the exponent law

$\begin{gathered} (2^x)^2+2^x\cdot2^1-63=0 \\ \end{gathered}$

Rewrite the equation 2^x = u

$\begin{gathered} u^2+u\cdot2-63=0 \\ \text{solve} \\ (u-7)(u+9)=0 \\ so \\ u=7 \\ \text{and } \\ u=-9 \end{gathered}$

then solve u

$\begin{gathered} 2^x=7 \\ x\ln (2)=\ln (7)^{} \\ x=\frac{\ln (7)}{\ln (2)} \\ \text{and} \\ 2^x=-9\text{ no solution} \end{gathered}$

Therefore:

$x=\frac{\ln (7)}{\ln (2)}$

Answer1: A. the solution set is

$\frac{\ln (7)}{\ln (2)}$

Answer2: A. the solution set is

$2.807$