If $a = b^x$, $b = c^y$ and $c = a^z$, then the value of $xyz$ is equal to

### Concept & Strategy This cyclic equation problem can be solved either by continuous algebraic substitution of the bases or by converting the exponential equations into logarithmic forms and multiplying them together. The substitution method is usually faster. Power of a power rule: $$ (x^m)^n = x^{mn} $$ ### Step-by-Step Solution * **Given:** Three interlocking exponential equations: $a = b^x$, $b = c^y$, and $c = a^z$. * **Calculation (Substitution Method):** Start with the first equation. $$ a = b^x $$ * Substitute the value of $b$ from the second equation ($b = c^y$) into the first equation. $$ a = (c^y)^x = c^{xy} $$ * Now, substitute the value of $c$ from the third equation ($c = a^z$) into the new equation. $$ a = (a^z)^{xy} $$ * Apply the power rule to multiply the exponents. $$ a^1 = a^{xyz} $$ * Since the bases on both sides of the equation are the same ($a$), their exponents must be equal. $$ 1 = xyz $$ ### Exam Strategy & Shortcut An alternative rapid method relies on logarithms. Convert each equation: $x = \frac{\log a}{\log b}$ $y = \frac{\log b}{\log c}$ $z = \frac{\log c}{\log a}$ When you multiply $xyz$, all the numerators and denominators perfectly cross-cancel: $\left(\frac{\log a}{\log b}\right) \times \left(\frac{\log b}{\log c}\right) \times \left(\frac{\log c}{\log a}\right) = 1$. This mental cross-cancellation allows you to see the answer instantly. ### Common Pitfall When substituting, students sometimes mistakenly add the exponents instead of multiplying them (e.g., changing $(c^y)^x$ to $c^{y+x}$). Remember that an exponent raised to another exponent means you multiply the powers. ### Final Answer **Therefore, the correct answer is 1.**