If $p^3 = q^4 = r^5 = s^6$, then the value of $\log_s(pqr)$ is equal to

Given: $p^3 = q^4 = r^5 = s^6$

To find: $\log_s(pqr)$

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Since all four expressions are equal, let's call this common value $k$.

$p^3 = q^4$

$= r^5$

$= s^6 = k$

This gives us a way to express each variable in terms of the same base, making calculations much easier.

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From the equations above:

$p^3 = k \rightarrow p = k^{1/3}$

$q^4 = k \rightarrow q = k^{1/4}$

$r^5 = k \rightarrow r = k^{1/5}$

$s^6 = k \rightarrow s = k^{1/6}$

If $x^n = k$, then $x = k^{1/n}$ because $(k^{1/n})^n = k$.

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$pqr = k^{1/3} \cdot k^{1/4} \cdot k^{1/5}$

Using the rule $a^m \cdot a^n = a^{m+n}$:

$pqr = k^{1/3 + 1/4 + 1/5}$

To add these fractions, we need a common denominator. The LCM of 3, 4, and 5 is 60.

$\tfrac{1}{3} = \tfrac{20}{60}$, $\tfrac{1}{4} = \tfrac{15}{60}$, $\tfrac{1}{5} = \tfrac{12}{60}$

$\tfrac{1}{3} + \tfrac{1}{4} + \tfrac{1}{5} = \tfrac{20 + 15 + 12}{60} = \tfrac{47}{60}$

Therefore: $pqr = k^{47/60}$

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$\log_s(pqr) = \log_{k^{1/6}}(k^{47/60})$

Using the logarithm property $\log_{a^m}(b^n) = \tfrac{n}{m} \log_a(b)$:

$\log_{k^{1/6}}(k^{47/60}) = \tfrac{47/60}{1/6} \log_k(k)$

$\tfrac{47/60}{1/6} = \tfrac{47}{60} \times \tfrac{6}{1} = \tfrac{47 \times 6}{60} = \tfrac{47}{10}$

Since $\log_k(k) = 1$:

$\log_s(pqr) = \tfrac{47}{10} \times 1 = \tfrac{47}{10}$

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$\log_s(pqr) = \tfrac{47}{10}$

This method works because we converted everything to the same base $k$, which allowed us to use logarithm properties effectively. The key insight is recognizing that when multiple expressions are equal, setting them equal to a common variable simplifies the entire problem.