Question

question 6 of 10
what is the solution to the equation below? round your answer to two decimal places.
$5 + 6 \cdot \log_{2} x = 14$

a. $x = 1.62$
b. $x = 2.83$
c. $x = 2.42$
d. $x = 2.25$

Explanation:

Step1: Isolate the logarithmic term

Subtract 5 from both sides of the equation \(5 + 6\cdot\log_{2}x = 14\).
\(6\cdot\log_{2}x = 14 - 5\)
\(6\cdot\log_{2}x = 9\)

Step2: Solve for the logarithmic term

Divide both sides by 6.
\(\log_{2}x=\frac{9}{6}\)
\(\log_{2}x = 1.5\)

Step3: Convert logarithmic to exponential form

Using the definition of a logarithm, if \(\log_{b}a = c\), then \(a = b^{c}\). Here, \(b = 2\), \(c = 1.5\), so:
\(x = 2^{1.5}\)

Step4: Calculate the value of \(2^{1.5}\)

\(2^{1.5}=2^{\frac{3}{2}}=\sqrt{2^{3}}=\sqrt{8}\approx2.83\)

Answer:

B. \(x = 2.83\)