use properties of logarithms to expand the logarithmic expression as much as possible. evaluate logarithmic…

use properties of logarithms to expand the logarithmic expression as much as possible. evaluate logarithmic expressions without using a calculator if possible. \n\\(\\log_{4} \\sqrt3{\frac{a^{6}b}{16}}\\)\n\\(\\log_{4} \\sqrt3{\frac{a^{6}b}{16}} = \\square\\)\n(use integers or fractions for any numbers in the expression.)

use properties of logarithms to expand the logarithmic expression as much as possible. evaluate logarithmic expressions without using a calculator if possible. \n\\(\\log_{4} \\sqrt3{\frac{a^{6}b}{16}}\\)\n\\(\\log_{4} \\sqrt3{\frac{a^{6}b}{16}} = \\square\\)\n(use integers or fractions for any numbers in the expression.)

Answer

Explanation:

Step1: Rewrite the radical as an exponent

The cube root can be written as an exponent of $\frac{1}{3}$, so we have $\log_{4}\left(\frac{a^{6}b}{16}\right)^{\frac{1}{3}}$.

Step2: Apply the power rule of logarithms

The power rule states that $\log_b(M^n) = n\log_b(M)$, so this becomes $\frac{1}{3}\log_{4}\left(\frac{a^{6}b}{16}\right)$.

Step3: Apply the quotient rule of logarithms

The quotient rule is $\log_b\left(\frac{M}{N}\right)=\log_b(M)-\log_b(N)$, so we get $\frac{1}{3}\left(\log_{4}(a^{6}b)-\log_{4}(16)\right)$.

Step4: Apply the product rule of logarithms

The product rule is $\log_b(MN)=\log_b(M)+\log_b(N)$, so this becomes $\frac{1}{3}\left(\log_{4}(a^{6})+\log_{4}(b)-\log_{4}(16)\right)$.

Step5: Apply the power rule again

For $\log_{4}(a^{6})$, using the power rule gives $6\log_{4}(a)$, and $\log_{4}(16)=\log_{4}(4^{2}) = 2$ (since $\log_b(b^x)=x$).

Step6: Simplify the expression

Substitute these back in: $\frac{1}{3}\left(6\log_{4}(a)+\log_{4}(b)-2\right)=\frac{1}{3}\times6\log_{4}(a)+\frac{1}{3}\log_{4}(b)-\frac{1}{3}\times2 = 2\log_{4}(a)+\frac{1}{3}\log_{4}(b)-\frac{2}{3}$.

Answer:

$2\log_{4}a+\frac{1}{3}\log_{4}b - \frac{2}{3}$