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Solve the following logarithm problem for the positive solution for
x.

log_(16)x=-(5)/(4)
Answer:

Solve the following logarithm problem for the positive solution for $x$ . $\newline$ $\log _{16} x=-\frac{5}{4}$ $\newline$ Answer:

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Quotient property of logarithms

Full solution

Q. Solve the following logarithm problem for the positive solution for

x

.

\newline

\log _{16} x=-\frac{5}{4}

\newline

Answer:

Rewrite in Exponential Form: We are given the logarithmic equation $\log_{16}(x) = -\left(\frac{5}{4}\right)$ . To solve for x, we need to rewrite the logarithmic equation in exponential form. $\newline$ The exponential form of a logarithm is given by: if $\log_b(a) = c$ , then $b^c = a$ . $\newline$ Therefore, we can rewrite our equation as $16^{-\left(\frac{5}{4}\right)} = x$ .
Calculate $16^{-(5/4)}$ : Now we need to calculate $16^{-(5/4)}$ . Since $16$ is $2$ raised to the $4$ th power ( $16 = 2^4$ ), we can rewrite the equation as $(2^4)^{-(5/4)}$ . Using the power of a power rule $(a^{m})^{n} = a^{m*n}$ , we get $2^{4*(-(5/4))}$ .
Calculate $2^{-5}$ : Calculating the exponent, we have $4*(-(5/4))$ which simplifies to $-5$ . So, we have $2^{-5} = x$ .
Final Result: Now we calculate $2^{-5}$ . $2^{-5}$ is the reciprocal of $2^5$ , which is $1/(2^5)$ . $2^5$ is $32$ , so $2^{-5}$ is $1/32$ . Therefore, $x = 1/32$ .

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