If $3^x = 5^y = 45^z$, then

### Concept & Strategy This is a classic "Common Constant" problem. When you have a chain of different bases equal to each other, introduce an arbitrary constant $k$. Express each base in terms of $k$, and then use the prime factorization of the largest base to link everything together into a single equation. ### Step-by-Step Solution * **Step 1:** Set the entire equality chain equal to a constant $k$. $$ 3^x = 5^y = 45^z = k $$ * **Step 2:** Isolate the bases ($3, 5, 45$) by raising both sides to reciprocal fractional powers. $ 3 = k^{\frac{1}{x}} $ $ 5 = k^{\frac{1}{y}} $ $ 45 = k^{\frac{1}{z}} $ * **Step 3:** Find the multiplicative relationship between the base numbers. We know that $ 45 = 9 \times 5 = 3^2 \times 5 $. * **Step 4:** Substitute the $k$-expressions into this relationship. $$ k^{\frac{1}{z}} = (k^{\frac{1}{x}})^2 \cdot k^{\frac{1}{y}} $$ * **Step 5:** Simplify the exponents using the rules $(a^m)^n = a^{mn}$ and $a^m \cdot a^n = a^{m+n}$. $$ k^{\frac{1}{z}} = k^{\frac{2}{x}} \cdot k^{\frac{1}{y}} $$ $$ k^{\frac{1}{z}} = k^{\left(\frac{2}{x} + \frac{1}{y}\right)} $$ * **Step 6:** Equate the exponents, as the base $k$ is the same. $$ \frac{1}{z} = \frac{2}{x} + \frac{1}{y} $$ * **Step 7:** Rearrange to match the given options. Move $\frac{1}{y}$ to the other side: $$ \frac{2}{x} = \frac{1}{z} - \frac{1}{y} $$ ### Exam Strategy & Shortcut **Prime Factorization Mapping:** Once you recognize the pattern $A^x = B^y = C^z$, look for the equation linking $A, B$, and $C$. Here, $3^2 \cdot 5^1 = 45^1$. The shortcut is to directly swap the base's exponent with the reciprocal of its variable: Power of 3 is $2 \rightarrow \frac{2}{x}$ Power of 5 is $1 \rightarrow \frac{1}{y}$ Power of 45 is $1 \rightarrow \frac{1}{z}$ So, $\frac{2}{x} + \frac{1}{y} = \frac{1}{z}$. Rearranging gives $\frac{2}{x} = \frac{1}{z} - \frac{1}{y}$. You can bypass the constant $k$ entirely on the exam. ### Common Pitfall The most common trap is algebraically mismanaging the exponent rules. Students often mistakenly write $k^{1/x \cdot 2} = k^{1/2x}$ instead of $k^{2/x}$. Remember that $(a^b)^c = a^{bc}$, which means multiplying the exponent by an integer puts the integer in the numerator. ### Final Answer Therefore, the correct answer is **$\frac{2}{x} = \frac{1}{z} - \frac{1}{y}$**.