Question

employees of a publishing company received an increase in salary of 8% plus a bonus of $1,200. let s(x) = 1.08x + 1200 represent the new salary in terms of the previous salary, x. find and interpret s(15,000).
$28,000; if an employees old salary was $28,000, then his/her new salary was $15,000 after the increase and bonus.
$12,788; if an employees old salary was $12,788, then his/her new salary was $15,000 after the increase and bonus.
$16,200; if an employees old salary was $15,000, then his/her new salary was $16,200 after the increase and bonus.
$17,400; if an employees old salary was $15,000, then his/her new salary was $17,400 after the increase and bonus.
question 34 (2 points)
given log_b(2) = 0.3562 and log_b(8) = 1.0686, evaluate log_b(16).
1.3868
-0.1147
1.4248
1.5395

Explanation:

First Sub - Question (Finding \(S(15,000)\))

Step 1: Substitute \(x = 15000\) into \(S(x)\)

We are given the function \(S(x)=1.08x + 1200\). To find \(S(15000)\), we substitute \(x = 15000\) into the function. So we have \(S(15000)=1.08\times15000+1200\).

Step 2: Calculate \(1.08\times15000\)

First, calculate \(1.08\times15000\). \(1.08\times15000=(1 + 0.08)\times15000=15000+0.08\times15000=15000 + 1200=16200\).

Step 3: Add the bonus

Then we add the bonus of \(1200\) to the result of \(1.08\times15000\). But wait, in the function \(S(x)=1.08x + 1200\), when \(x = 15000\), \(1.08\times15000=16200\), and then \(16200+1200 = 17400\)? Wait, no, wait. Wait, \(1.08\times15000=16200\), then \(S(15000)=1.08\times15000 + 1200=16200+1200=17400\). Wait, let's re - calculate:

\(1.08\times15000\): \(15000\times1=15000\), \(15000\times0.08 = 1200\), so \(15000 + 1200=16200\). Then \(16200+1200 = 17400\).

The interpretation is: If an employee's old salary (\(x\)) was \(\$15,000\), then the new salary \(S(x)\) after an \(8\%\) increase (\(1.08x\)) and a bonus of \(\$1200\) is \(\$17,400\).

Second Sub - Question (Evaluating \(\log_{b}(16)\))

Step 1: Express \(16\) as a product of known values

We know that \(16=2\times8\). By the property of logarithms \(\log_{b}(mn)=\log_{b}(m)+\log_{b}(n)\) (where \(m = 2\) and \(n = 8\) and \(m,n>0\) and \(b>0,b
eq1\)).

Step 2: Substitute the known values of logarithms

We are given that \(\log_{b}(2)=0.3562\) and \(\log_{b}(8)=1.0686\). So \(\log_{b}(16)=\log_{b}(2\times8)=\log_{b}(2)+\log_{b}(8)\).

Step 3: Calculate the sum

Substitute the values: \(\log_{b}(2)+\log_{b}(8)=0.3562 + 1.0686=1.4248\)? Wait, no, wait \(0.3562+1.0686 = 1.4248\)? Wait, \(0.3562+1.0686=1.4248\)? Wait, \(0.3562+1.0686 = 1.4248\). But wait, another way: We know that \(16 = 2^{4}\), and \(\log_{b}(a^{n})=n\log_{b}(a)\). Since \(\log_{b}(2) = 0.3562\), then \(\log_{b}(16)=\log_{b}(2^{4})=4\log_{b}(2)=4\times0.3562 = 1.4248\).

First Sub - Question Answer:

\(S(15000) = 17400\); If an employee's old salary was \(\$15,000\), then his/her new salary was \(\$17,400\) after the increase and bonus. So the correct option is "D. \(\$17,400\); If an employee's old salary was \(\$15,000\), then his/her new salary was \(\$17,400\) after the increase and bonus." (assuming the options are labeled A - D in order)

Second Sub - Question Answer:

Using the property \(\log_{b}(16)=\log_{b}(2\times8)=\log_{b}(2)+\log_{b}(8)\) or \(\log_{b}(16)=\log_{b}(2^{4}) = 4\log_{b}(2)\). Using \(\log_{b}(2)=0.3562\), \(4\times0.3562 = 1.4248\). So the correct option is "C. \(1.4248\)" (assuming the options are labeled in order)

For the first sub - question (salary function):

Answer:

\(\$17,400\); If an employee's old salary was \(\$15,000\), then his/her new salary was \(\$17,400\) after the increase and bonus.

For the second sub - question (logarithm):