Answer:
Value of expression in single logarithm is [tex]\log_3\left(2z^2\right)[/tex].
Step-by-step explanation:
Given expression is,
[tex]2\left(\log_3\left(8\right)+\log_3\left(z\right)\right)-\log_3\left(3^4-7^2\right)[/tex]
Now using logarithmic rule to solve the expression as follows,
Applying product rule of logarithmic,
[tex]\log_c\left(a\right)+\log_c\left(b\right)=\log_c\left(ab\right)[/tex]
Therefore,
[tex]2\log_3\left(8z\right)-\log_3\left(3^4-7^2\right)[/tex]
Applying power rule of logarithmic,
[tex]a\log_c\left(b\right)=\log_c\left(b^a\right)[/tex]
Therefore,
[tex]\log_3\left(\left(8z\right)^2\right)-\log_3\left(3^4-7^2\right)[/tex]
[tex]\log_3\left(\left(64z^2\right)\right)-\log_3\left(3^4-7^2\right)[/tex]
Applying quotient rule of logarithmic,
[tex]\log_c\left(a\right)-\log_c\left(b\right)=\log_c\left(\frac{a}{b}\right)[/tex]
Therefore,
[tex]\log_3\left(\dfrac{\left(64z^2\right)^2}{3^4-7^2}\right)[/tex]
Simplifying,
[tex]\log_3\left(\dfrac{\left(64z^2\right)^2}{81-49}\right)[/tex]
[tex]\log_3\left(\dfrac{\left(64z^2\right)^2}{32}\right)[/tex]
[tex]\log_3\left(2z^2\right)[/tex]
Therefore value of expression is [tex]\log_3\left(2z^2\right)[/tex]
