`"If" ("log"_(10)a)/(2) = ("log"_(10)b)/(3) = ("log"_(10)c)/(5)`, then bc =

Options

A. a
B. `a^(2)`
C. `a^(3)`
D. `a^(4)`

Explanation

To solve the problem where we have the equation:

\[ \frac{\log_{10} a}{2} = \frac{\log_{10} b}{3} = \frac{\log_{10} c}{5} \]

we will denote this common value as \( k \). Therefore, we can rewrite the equations as follows:

1. \(\frac{\log_{10} a}{2} = k\)
2. \(\frac{\log_{10} b}{3} = k\)
3. \(\frac{\log_{10} c}{5} = k\)

### Step 1: Express \( a \), \( b \), and \( c \) in terms of \( k \)

From the first equation, we can express \( \log_{10} a \):

\[ \log_{10} a = 2k \]

Using the definition of logarithms, we can express \( a \):

\[ a = 10^{2k} \]

From the second equation, we can express \( \log_{10} b \):

\[ \log_{10} b = 3k \]

Thus, we can express \( b \):

\[ b = 10^{3k} \]

From the third equation, we can express \( \log_{10} c \):

\[ \log_{10} c = 5k \]

Thus, we can express \( c \):

\[ c = 10^{5k} \]

### Step 2: Find the product \( bc \)

Now, we need to find the product \( bc \):

\[ bc = b \cdot c = (10^{3k}) \cdot (10^{5k}) \]

Using the property of exponents that states \( a^m \cdot a^n = a^{m+n} \):

\[ bc = 10^{3k + 5k} = 10^{8k} \]

### Step 3: Express \( bc \) in terms of \( a \)

We already know that \( a = 10^{2k} \). To express \( 10^{8k} \) in terms of \( a \), we can manipulate the expression for \( a \):

\[ a = 10^{2k} \implies 10^{8k} = (10^{2k})^4 = a^4 \]

### Conclusion

Thus, we find that:

\[ bc = a^4 \]

### Final Answer

The value of \( bc \) is \( a^4 \).

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