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assignment 6.4: logarithmic equations and ineq
score: 2/19 answered: 2/16
question 3
if \\(\log_{x} \left( \dfrac{1}{64} \
ight) = -\dfrac{3}{2}\\), then
\\(x = \square\\)
question help: message instructor

Explanation:

Step1: Convert log to exponential form

Recall $\log_b(a)=c \implies b^c=a$. So:
$$x^{-\frac{3}{2}} = \frac{1}{64}$$

Step2: Rewrite negative exponent

A negative exponent means reciprocal:
$$\frac{1}{x^{\frac{3}{2}}} = \frac{1}{64}$$

Step3: Cross-multiply to simplify

$$x^{\frac{3}{2}} = 64$$

Step4: Rewrite $x^{\frac{3}{2}}$ as $(\sqrt{x})^3$

$$(\sqrt{x})^3 = 64$$

Step5: Take cube root of both sides

$$\sqrt{x} = 64^{\frac{1}{3}} = 4$$

Step6: Square both sides to solve for x

$$x = 4^2 = 16$$

Answer:

$16$