Answer:
[tex]\displaystyle log_\frac{1}{2}(64)=-6[/tex]
Step-by-step explanation:
Properties of Logarithms
We'll recall below the basic properties of logarithms:
[tex]log_b(1) = 0[/tex]
Logarithm of the base:
[tex]log_b(b) = 1[/tex]
Product rule:
[tex]log_b(xy) = log_b(x) + log_b(y)[/tex]
Division rule:
[tex]\displaystyle log_b(\frac{x}{y}) = log_b(x) - log_b(y)[/tex]
Power rule:
[tex]log_b(x^n) = n\cdot log_b(x)[/tex]
Change of base:
[tex]\displaystyle log_b(x) = \frac{ log_a(x)}{log_a(b)}[/tex]
Simplifying logarithms often requires the application of one or more of the above properties.
Simplify
[tex]\displaystyle log_\frac{1}{2}(64)[/tex]
Factoring [tex]64=2^6[/tex].
[tex]\displaystyle log_\frac{1}{2}(64)=\displaystyle log_\frac{1}{2}(2^6)[/tex]
Applying the power rule:
[tex]\displaystyle log_\frac{1}{2}(64)=6\cdot log_\frac{1}{2}(2)[/tex]
Since
[tex]\displaystyle 2=(1/2)^{-1}[/tex]
[tex]\displaystyle log_\frac{1}{2}(64)=6\cdot log_\frac{1}{2}((1/2)^{-1})[/tex]
Applying the power rule:
[tex]\displaystyle log_\frac{1}{2}(64)=-6\cdot log_\frac{1}{2}(\frac{1}{2})[/tex]
Applying the logarithm of the base:
[tex]\mathbf{\displaystyle log_\frac{1}{2}(64)=-6}[/tex]
