Answer :
To solve for the absolute value of [tex](xy + 2z)[/tex], given the equations:
- [tex]x = \log_2 3 - 4\log_5 \left(\frac{3}{4}\right)[/tex],
- [tex]y = \log_5 \left(\frac{5}{4}\right)[/tex],
- [tex]z = \log_2 \left(\log_{(5/4)} e\right)[/tex],
we'll break it down step by step.
Step 1: Simplify Expression for x
The expression for [tex]x[/tex] is:
[tex]x = \log_2 3 - 4\log_5 \left(\frac{3}{4}\right)[/tex]
Using the logarithm property [tex]a\log_b c = \log_b c^a[/tex], we can rewrite:
[tex]x = \log_2 3 - \log_5 \left(\frac{3}{4}\right)^4[/tex]
Step 2: Simplify Expression for y
The expression for [tex]y[/tex] is already straightforward:
[tex]y = \log_5 \left(\frac{5}{4}\right)[/tex]
Step 3: Simplify Expression for z
Given [tex]z = \log_2 \left(\log_{(5/4)} e\right)[/tex], apply the change of base formula:
[tex]\log_{(5/4)} e = \frac{\log e}{\log (5/4)} = \frac{1}{\log (5/4)}[/tex]
Then,
[tex]z = \log_2 \left(\frac{1}{\log (5/4)}\right) = -\log_2 (\log(5/4))[/tex]
Step 4: Calculate [tex]xy[/tex]
Using the known expressions for [tex]x[/tex] and [tex]y[/tex]:
- Substitute and simplify both expressions as applicable to compute [tex]xy[/tex].
Step 5: Calculate [tex]2z[/tex]
Take [tex]2z[/tex], using
[tex]2z = -2\log_2 (\log (5/4))[/tex]
Step 6: Combine the Expressions
Now, compute [tex]xy + 2z[/tex]. Remember, the solution requires the absolute value:
[tex]|xy + 2z| = ?[/tex]
Taking these steps and solving will yield the exact numeric result. These calculations can be assisted further using a calculator if precise decimal values are needed.
The solution involves logarithm properties, change of base formula, and simplification techniques commonly found in college-level mathematics.