Problem Understanding: Understand the problem. We need to find the value of the logarithm of 1/2 with base 16. The logarithm function answers the question: ＂To what power must we raise the base (in this case, 16) to obtain the number inside the log (in this case, 1/2)?＂Logarithm Property: Recall the logarithm property that log base b of 1 is always 0. log_b(1) = 0 for any base b, because any number raised to the power of 0 is 1.Finding a Power of 16: Recognize that the problem cannot be directly solved using the property from Step 2 because the argument of the logarithm is 1/2, not 1. We need to find a way to express 1/2 as a power of 16.Expressing 1/2 as a Power of 16: Express 1/2 as a power of 16. Since 16 is 2 to the 4th power (16 = 2^4), we can try to express 1/2 in terms of a power of 2. 1/2 is the same as 2^(-1).Simplifying the Expression: Express 2^(-1) in terms of a power of 16. Since 16 is 2^4, we can write 2^(-1) as (2^4)^(-1/4).Mistake in Simplification: Simplify the expression. (2^4)^(-1/4) simplifies to 2^(-4/4), which is 2^(-1).Realize a mistake has been made in the simplification process. The simplification in Step 6 is incorrect because (2^4)^(-1/4) should simplify to 2^(4*(-1/4)), which is 2^(-1), but we already started with 2^(-1), so we have not changed the expression and are in a loop.

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$\log _{16} \frac{1}{2}=$

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Grade 6

Multiply using the distributive property

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\log _{16} \frac{1}{2}=

Problem Understanding: Understand the problem. $\newline$ We need to find the value of the logarithm of $\frac{1}{2}$ with base $16$ . The logarithm function answers the question: ＂To what power must we raise the base (in this case, $16$ ) to obtain the number inside the log (in this case, $\frac{1}{2}$ )?＂
Logarithm Property: Recall the logarithm property that $\log_b(1) = 0$ for any base $b$ , because any number raised to the power of $0$ is $1$ .
Finding a Power of $16$ : Recognize that the problem cannot be directly solved using the property from Step $2$ because the argument of the logarithm is $\frac{1}{2}$ , not $1$ . $\newline$ We need to find a way to express $\frac{1}{2}$ as a power of $16$ .
Expressing rac{ $1$ }{ $2$ } as a Power of $16$ : Express rac{ $1$ }{ $2$ } as a power of $16$ . $\newline$ Since $16$ is $2$ to the $4$ th power ( $16$ = $2$ ^{ $4$ }), we can try to express rac{ $1$ }{ $2$ } in terms of a power of $2$ . $\newline$ rac{ $1$ }{ $2$ } is the same as $2$ ^{ $-1$ }.
Simplifying the Expression: Express $2^{-1}$ in terms of a power of $16$ . $\newline$ Since $16$ is $2^4$ , we can write $2^{-1}$ as $(2^4)^{-\frac{1}{4}}$ .
Mistake in Simplification: Simplify the expression. $\newline$ $(2^4)^{-\frac{1}{4}}$ simplifies to $2^{-\frac{4}{4}}$ , which is $2^{-1}$ .
Mistake in Simplification: Simplify the expression. $\newline$ $(2^4)^{-\frac{1}{4}}$ simplifies to $2^{-\frac{4}{4}}$ , which is $2^{-1}$ .Realize a mistake has been made in the simplification process. $\newline$ The simplification in Step $6$ is incorrect because $(2^4)^{-\frac{1}{4}}$ should simplify to $2^{4*(-\frac{1}{4})}$ , which is $2^{-1}$ , but we already started with $2^{-1}$ , so we have not changed the expression and are in a loop.