Question

calculator to obtain a decimal approximation for the solution.
2^{x + 2}=345
the solution set expressed in terms of logarithms is \boxed{}.
(use a comma to separate answers as needed. simplify your answer. use integers or fractions for any numbers in the
in for natural logarithm and \log for common logarithm.)

Explanation:

Step1: Take log on both sides

Take the common logarithm (log base 10) of both sides of the equation \(2^{x + 2}=345\). Using the logarithm power rule \(\log(a^b)=b\log(a)\), we get:
\(\log(2^{x + 2})=\log(345)\)
\((x + 2)\log(2)=\log(345)\)

Step2: Solve for x

First, divide both sides by \(\log(2)\):
\(x + 2=\frac{\log(345)}{\log(2)}\)
Then, subtract 2 from both sides to solve for \(x\):
\(x=\frac{\log(345)}{\log(2)}-2\)

If we use natural logarithm (ln), the steps are similar:
Take the natural logarithm of both sides: \(\ln(2^{x + 2})=\ln(345)\)
Using the power rule \(\ln(a^b)=b\ln(a)\), we have \((x + 2)\ln(2)=\ln(345)\)
Divide by \(\ln(2)\): \(x + 2=\frac{\ln(345)}{\ln(2)}\)
Subtract 2: \(x=\frac{\ln(345)}{\ln(2)}-2\)

Now, calculate the decimal approximation. We know that \(\log(345)\approx2.5378\), \(\log(2)\approx0.3010\), so \(\frac{\log(345)}{\log(2)}\approx\frac{2.5378}{0.3010}\approx8.431\), then \(x\approx8.431 - 2 = 6.431\).

Or using natural logarithm: \(\ln(345)\approx5.842\), \(\ln(2)\approx0.6931\), \(\frac{\ln(345)}{\ln(2)}\approx\frac{5.842}{0.6931}\approx8.431\), then \(x\approx8.431 - 2 = 6.431\).

Answer:

In terms of logarithms, \(x=\frac{\log(345)}{\log(2)}-2\) (or \(x=\frac{\ln(345)}{\ln(2)}-2\)); the decimal approximation is approximately \(6.43\) (rounded to two decimal places) or more precisely \(6.431\) (depending on calculation precision).