Chapter 9 - Section 9.4 - Properties of Logarithms - Exercise Set: 23

$\frac{1}{2}\log_4{x} - 3$

Work Step by Step

RECALL: (1) $\log{(b^c)}=c \cdot \log{b}$ (2) $\log_b{(xy)} = \log_b{x} + \log_b{y}$ (3) $\log_b{(\frac{x}{y})}=\log_b{x} - \log_b{y}$ Use rule (3) above to obtain: $=log_4{\sqrt{x}}-\log_4{64}$ Note that $\sqrt{x} = x^{\frac{1}{2}}$ and $64=4^3$. Thus, the expression above is equivalent to: $=\log_4{(x^{\frac{1}{2}})}-\log_4{(4^3)}$ Use rule (1) above to obtain: $=\frac{1}{2}\log_4{x} - 3\log_4{4}$ Use the rule $\log_b{b} = 1$ to obtain: $=\frac{1}{2}\log_4{x} - 3 \cdot 1 \\=\frac{1}{2}\log_4{x} - 3$

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