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

log_(x)((1)/(4))=-(2)/(5)
Answer:

Solve the following logarithm problem for the positive solution for $x$ . $\newline$ $\log _{x} \frac{1}{4}=-\frac{2}{5}$ $\newline$ 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 _{x} \frac{1}{4}=-\frac{2}{5}

\newline

Answer:

Understand the logarithmic equation: Understand the logarithmic equation. $\newline$ The given logarithmic equation is $\log_x(\frac{1}{4}) = -\frac{2}{5}$ . This means that $x$ raised to the power of $-\frac{2}{5}$ equals $\frac{1}{4}$ .
Convert to exponential form: Convert the logarithmic form to exponential form. $\newline$ Using the definition of a logarithm, we can rewrite the equation in its exponential form: $x^{(-2/5)} = \frac{1}{4}$ .
Solve for x: Solve for x. $\newline$ Since $x^{(-2/5)} = \frac{1}{4}$ , we can take both sides to the power of $-\frac{5}{2}$ to isolate $x$ . Doing so, we get: $\newline$ $(x^{(-2/5)})^{(-5/2)} = (\frac{1}{4})^{(-5/2)}$
Simplify the equation: Simplify the equation. $\newline$ When we raise a power to a power, we multiply the exponents. Therefore, we have: $\newline$ $x^{(-\frac{2}{5})\cdot(-\frac{5}{2})} = (\frac{1}{4})^{(-\frac{5}{2})}$ $\newline$ $x^1 = (\frac{1}{4})^{(-\frac{5}{2})}$
Calculate the right side: Calculate the right side of the equation. $\newline$ To calculate $(1/4)^{(-5/2)}$ , we can first take the reciprocal of $1/4$ , which is $4$ , and then raise it to the power of $5/2$ : $\newline$ $(1/4)^{(-5/2)} = 4^{(5/2)}$
Find square root of $4$ : Find the square root of $4$ and then raise it to the $5$ th power. $\newline$ The square root of $4$ is $2$ , so we have: $\newline$ $4^{5/2} = (2^2)^{5/2} = 2^{2*(5/2)} = 2^5$
Calculate $2^5$ : Calculate $2^5$ . $2^5$ is $2$ multiplied by itself $5$ times, which equals $32$ . So, $x = 32$ .

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