To find the approximate value of [tex]\( q \)[/tex] in the equation [tex]\( q + \log_2 6 = 2q + 2 \)[/tex], let’s go through the steps of solving the equation.
1. Start with the given equation:
[tex]\[
q + \log_2 6 = 2q + 2
\][/tex]
2. Isolate [tex]\( q \)[/tex]:
Subtract [tex]\( q \)[/tex] from both sides of the equation:
[tex]\[
\log_2 6 = q + 2
\][/tex]
3. Isolate [tex]\( q \)[/tex] further:
Subtract 2 from both sides:
[tex]\[
\log_2 6 - 2 = q
\][/tex]
4. Simplify the expression:
We need to calculate [tex]\( \log_2 6 - 2 \)[/tex].
The change of base formula for logarithms can be used here:
[tex]\[
\log_2 6 = \frac{\log 6}{\log 2}
\][/tex]
Thus,
[tex]\[
q = \frac{\log 6}{\log 2} - 2
\][/tex]
5. Express the logarithm in base 2 using simpler forms:
We can rewrite [tex]\( \log 6 \)[/tex] as [tex]\( \log (2 \cdot 3) = \log 2 + \log 3 \)[/tex].
Therefore:
[tex]\[
\frac{\log 6}{\log 2} = \frac{\log 2 + \log 3}{\log 2} = 1 + \frac{\log 3}{\log 2}
\][/tex]
So,
[tex]\[
q = 1 + \frac{\log 3}{\log 2} - 2 = -1 + \frac{\log 3}{\log 2}
\][/tex]
6. Identify the approximate numerical value:
The result reduces to:
[tex]\[
q = -1 + \frac{\log 3}{\log 2}
\][/tex]
The approximate value of [tex]\(\frac{\log 3}{\log 2}\)[/tex] is about 1.585 (as [tex]\( \log_2 3 \approx 1.585 \)[/tex]).
Substituting this back:
[tex]\[
q \approx -1 + 1.585 = 0.585
\][/tex]
7. Match the calculated value with the given options:
Among the provided options:
- [tex]\( -1.613 \)[/tex]
- [tex]\( -1.522 \)[/tex]
- [tex]\( 0.585 \)[/tex]
- [tex]\( 3.079 \)[/tex]
The value [tex]\( q \approx 0.585 \)[/tex] matches with our calculated result. Thus, the approximate value of [tex]\( q \)[/tex] is:
[tex]\[
\boxed{0.585}
\][/tex]
