Question

2
convert each exponential expression to an logarithmic expression

a) $6^3 = 216$

b) $7^5 = 16,807$

3
solve for x

a) $\log_{64}(x) = \frac{1}{3}$

b) $\log_{x}343 = 3$

Explanation:

For Question 2 (Exponential to Logarithmic Conversion)

Use the relationship $b^y = x \iff \log_b(x) = y$ for each conversion.

Step1: Apply log-exponential relation

For $\log_{64}(x) = \frac{1}{3}$, use $\log_b(x)=y \implies x=b^y$.
$x = 64^{\frac{1}{3}}$

Step2: Simplify the exponential term

Rewrite 64 as $4^3$.
$x = (4^3)^{\frac{1}{3}} = 4^{3 \times \frac{1}{3}} = 4$

Step3: Apply log-exponential relation (part b)

For $\log_{x}(343) = 3$, use $\log_b(x)=y \implies b^y=x$.
$x^3 = 343$

Step4: Solve for x (part b)

Rewrite 343 as $7^3$, then take cube root.
$x = \sqrt[3]{343} = \sqrt[3]{7^3} = 7$

Answer:

a) $\log_{6}(216) = 3$
b) $\log_{7}(16807) = 5$

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For Question 3 (Solve for x)