Sure, let's go through the steps to solve the equation [tex]\( 9 \cdot e^{2z} = 54 \)[/tex] for [tex]\( z \)[/tex].
1. Divide both sides by 9:
[tex]\[
e^{2z} = \frac{54}{9}
\][/tex]
Simplifying, we get:
[tex]\[
e^{2z} = 6
\][/tex]
2. Take the natural logarithm (ln) of both sides:
Taking the natural logarithm (ln) on both sides will help us isolate the exponent. This gives us:
[tex]\[
\ln(e^{2z}) = \ln(6)
\][/tex]
By the properties of logarithms, specifically [tex]\(\ln(e^x) = x\)[/tex], we have:
[tex]\[
2z = \ln(6)
\][/tex]
3. Solve for [tex]\( z \)[/tex]:
Divide both sides by 2 to isolate [tex]\( z \)[/tex]:
[tex]\[
z = \frac{\ln(6)}{2}
\][/tex]
Therefore, the solution for [tex]\( z \)[/tex] in terms of natural logarithms is:
[tex]\[
z = \frac{\ln(6)}{2}
\][/tex]
Next, let's approximate the value of [tex]\( z \)[/tex] and round it to the nearest thousandth.
Using a calculator or a logarithm table:
[tex]\[
\ln(6) \approx 1.791759
\][/tex]
Thus:
[tex]\[
z \approx \frac{1.791759}{2} \approx 0.895880
\][/tex]
Rounding to the nearest thousandth, we get:
[tex]\[
z \approx 0.896
\][/tex]
So, the approximate value of [tex]\( z \)[/tex] rounded to the nearest thousandth is:
[tex]\[
z \approx 0.896
\][/tex]
To summarize:
[tex]\[
z = \frac{\ln(6)}{2}
\][/tex]
[tex]\[
z \approx 0.896
\][/tex]
