To solve the equation [tex]\( 1.5(4)^{2z} = 12 \)[/tex] for [tex]\( z \)[/tex], we can proceed with the following steps:
1. Isolate the exponential term:
Divide both sides of the equation by 1.5 to isolate the term involving [tex]\( z \)[/tex]:
[tex]\[
(4)^{2z} = \frac{12}{1.5}
\][/tex]
Simplify the fraction:
[tex]\[
(4)^{2z} = 8
\][/tex]
2. Apply the logarithm:
Take the natural logarithm (logarithm base [tex]\( e \)[/tex]) of both sides to solve for [tex]\( z \)[/tex]:
[tex]\[
\log((4)^{2z}) = \log(8)
\][/tex]
3. Use the power rule of logarithms:
The logarithm of an exponentiated term can be simplified using the power rule, which states [tex]\(\log(a^b) = b \log(a)\)[/tex]. Applying this rule, we get:
[tex]\[
2z \cdot \log(4) = \log(8)
\][/tex]
4. Solve for [tex]\( z \)[/tex]:
To solve for [tex]\( z \)[/tex], divide both sides by [tex]\( 2 \log(4) \)[/tex]:
[tex]\[
z = \frac{\log(8)}{2 \log(4)}
\][/tex]
5. Evaluate the logarithms:
Consider the values of the logarithms involved and calculate:
[tex]\[
\log(8) \text{ and } \log(4)
\][/tex]
6. Compute the fraction:
Calculate the value of [tex]\( z \)[/tex]:
[tex]\[
z \approx 0.75
\][/tex]
7. Round to the nearest hundredth:
Finally, the value of [tex]\( z \)[/tex], when rounded to the nearest hundredth, is 0.75.
So, the value of [tex]\( x \)[/tex] that satisfies the equation is [tex]\( 0.75 \)[/tex].
