If $\log_8 p = 25$ and $\log_2 q = 5$, then

Concept & Formula To relate variables from two different logarithmic equations, convert both to exponential form and express them using a common prime base so their exponents can be directly compared. $$ \log_a b = x \implies a^x = b $$ Step-by-Step Solution * **Given:** Two equations: $\log_8 p = 25$ and $\log_2 q = 5$. * **Calculation:** Convert both equations to exponential form: Equation 1: $8^{25} = p$ Equation 2: $2^5 = q$ * Notice that the bases $8$ and $2$ are related. Convert the base $8$ in Equation 1 to base $2$: $$ (2^3)^{25} = p $$ * Simplify the exponents using $(a^m)^n = a^{m \cdot n}$: $$ 2^{75} = p $$ * Now look at Equation 2 ($2^5 = q$). We need to raise this equation to a power that turns $2^5$ into $2^{75}$ to match $p$. * Since $75 \div 5 = 15$, raise both sides of Equation 2 to the power of $15$: $$ (2^5)^{15} = q^{15} $$ $$ 2^{75} = q^{15} $$ * Since both $p$ and $q^{15}$ are equal to $2^{75}$, we can equate them: $$ p = q^{15} $$ Exam Strategy & Shortcut Observe the bases: $8$ is $2^3$. This means $p$ grows $3$ times faster in the exponent. $3 \times 25 = 75$. $q$ is at exponent $5$. How many times does $5$ go into $75$? $15$ times. Therefore, $q$ must be raised to the $15$th power to catch up to $p$. Common Pitfall A typical mistake is trying to divide the logarithmic equations directly without converting to exponential form or aligning the bases, leading to incorrect algebraic relationships like $p = q^5$. Final Answer **Therefore, the correct answer is p = q^{15}.**