Question

for the exponential equation $2^{x + 3} = 37$, complete parts a) and b).
a. write the exact answer with natural logarithms.
b. approximate the solution by rounding to four decimal places.
a. select the correct choice and, if necessary, fill in the answer box to complete your choice.
a. the exact solution to the equation $2^{x + 3} = 37$ using natural logarithms is $x = \square$.
(type an exact answer in simplified form.)
b. there is no solution. the solution set is the empty set, $\varnothing$.
b. select the correct choice and, if necessary, fill in the answer box to complete your choice.
a. the approximate solution to the equation $2^{x + 3} = 37$ rounded to 4 decimal places is $x \approx \square$.
(round to four decimal places as needed.)
b. there is no solution. the solution set is the empty set, $\varnothing$.

Explanation:

Part a)

Step1: Take natural log of both sides

Take the natural logarithm of both sides of the equation \(2^{x + 3}=37\). Using the property \(\ln(a^b)=b\ln(a)\), we get \(\ln(2^{x + 3})=\ln(37)\), which simplifies to \((x + 3)\ln(2)=\ln(37)\).

Step2: Solve for x

Divide both sides by \(\ln(2)\): \(x + 3=\frac{\ln(37)}{\ln(2)}\). Then subtract 3 from both sides: \(x=\frac{\ln(37)}{\ln(2)}-3\).

Step1: Calculate the value

First, find the values of \(\ln(37)\) and \(\ln(2)\). \(\ln(37)\approx3.6109\) and \(\ln(2)\approx0.6931\). Then \(\frac{\ln(37)}{\ln(2)}\approx\frac{3.6109}{0.6931}\approx5.2097\).

Step2: Subtract 3

Subtract 3 from the result: \(x\approx5.2097 - 3=2.2097\).

Answer:

A. The exact solution to the equation \(2^{x + 3}=37\) using natural logarithms is \(x=\frac{\ln(37)}{\ln(2)}-3\).

Part b)