Consider the equation [tex]$9 \cdot e^{2z} = 54$[/tex].

1. Solve the equation for [tex]$z$[/tex]. Express the solution as a logarithm in base-e.
[tex]\[
z = \square
\][/tex]

2. Approximate the value of [tex]$z$[/tex]. Round your answer to the nearest thousandth.
[tex]\[
z \approx \square
\][/tex]



Answer :

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]