Question

solve the equation. give an exact solution, and also approximate the solution to four decimal places.
2^{x + 3}=5

write the exact solution.
x = 3 - \frac{\ln(5)}{\ln(2)} (simplify your answer.)

Explanation:

Step1: Take natural logarithm of both sides

$\ln(2^{x + 3})=\ln(5)$

Step2: Use logarithm property $\ln(a^b)=b\ln(a)$

$(x + 3)\ln(2)=\ln(5)$

Step3: Solve for $x$

$x+3=\frac{\ln(5)}{\ln(2)}$, then $x=\frac{\ln(5)}{\ln(2)}-3$

Answer:

Exact solution: $x=\frac{\ln(5)}{\ln(2)}-3$
Approximate solution:
First, calculate $\frac{\ln(5)}{\ln(2)}\approx2.3219$, then $x\approx2.3219 - 3=- 0.6781$