To solve these logarithmic equations, we will rewrite each equation in exponential form, using the property:

$\log_b(a) = c \implies b^c = a$

(a) $\log_x(8) = \frac{1}{2}$

Using the definition:

$x^{\frac{1}{2}} = 8$

This simplifies to:

$\sqrt{x} = 8$

Squaring both sides:

$x = 8^2 = 64$

So, the solution is:

$x = 64$

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(b) $\log_x(5) = \frac{1}{3}$

Using the definition:

$x^{\frac{1}{3}} = 5$

This simplifies to:

$\sqrt[3]{x} = 5$

Cubing both sides:

$x = 5^3 = 125$

So, the solution is:

$x = 125$

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Final Answers:

(a) $x = 64$
(b) $x = 125$

Let me know if you'd like further explanations!

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Here are some related questions to expand your understanding:

1. How does the definition of logarithms help rewrite exponential equations?
2. Can you verify the solutions $x = 64$ and $x = 125$ by substitution?
3. What is the meaning of fractional powers like $\frac{1}{2}$ and $\frac{1}{3}$ in terms of roots?
4. How would the solutions change if the bases of the logarithms were different?
5. Can you think of real-life applications where solving logarithmic equations is useful?

Tip: Always confirm logarithmic solutions by substituting them back into the original equation to verify correctness.