To solve for [tex]\( y \)[/tex] in the equation [tex]\( 7^{y-7} = 2 \)[/tex], let's follow a step-by-step approach.
1. Take the natural logarithm of both sides:
Given the equation [tex]\( 7^{y-7} = 2 \)[/tex], we take the natural logarithm (denoted as [tex]\(\ln\)[/tex]) on both sides:
[tex]\[
\ln(7^{y-7}) = \ln(2)
\][/tex]
2. Utilize the power rule of logarithms:
The power rule states that [tex]\(\ln(a^b) = b \ln(a)\)[/tex]. Applying this rule, we get:
[tex]\[
(y-7) \ln(7) = \ln(2)
\][/tex]
3. Solve for [tex]\( y \)[/tex]:
To isolate [tex]\( y \)[/tex], we need to divide both sides by [tex]\(\ln(7)\)[/tex]:
[tex]\[
y - 7 = \frac{\ln(2)}{\ln(7)}
\][/tex]
Now, we add 7 to both sides to solve for [tex]\( y \)[/tex]:
[tex]\[
y = \frac{\ln(2)}{\ln(7)} + 7
\][/tex]
4. Compute the natural logarithms:
The natural logarithm of 2 ([tex]\(\ln(2)\)[/tex]) is approximately 0.6931471805599453.
The natural logarithm of 7 ([tex]\(\ln(7)\)[/tex]) is approximately 1.9459101490553132.
5. Calculate the value of [tex]\( y \)[/tex]:
Substitute the computed logarithm values:
[tex]\[
y = \frac{0.6931471805599453}{1.9459101490553132} + 7
\][/tex]
First, compute the division:
[tex]\[
\frac{0.6931471805599453}{1.9459101490553132} \approx 0.356207187108022
\][/tex]
Then, add 7:
[tex]\[
y = 0.356207187108022 + 7 \approx 7.356207187108022
\][/tex]
6. Round to the nearest hundredth:
The value [tex]\( 7.356207187108022 \)[/tex] rounded to the nearest hundredth is 7.36.
Therefore, the value of [tex]\( y \)[/tex] is:
[tex]\[
y = 7.36
\][/tex]
