If `("log"a)/(3) = ("log"b)/(4) = ("log"c)/(5)`, then ca equals

Options

A. 2b
B. `b^(2)`
C. 8b
D. 4b

Answer

To solve the problem, we start with the given equation:

\[ \frac{\log a}{3} = \frac{\log b}{4} = \frac{\log c}{5} = k \]

### Step 1: Express logarithms in terms of \( k \)

From the equation, we can express the logarithms of \( a \), \( b \), and \( c \) in terms of \( k \):

\[ \log a = 3k \]
\[ \log b = 4k \]
\[ \log c = 5k \]

### Step 2: Convert logarithmic expressions to exponential form

Now, we can convert these logarithmic expressions into exponential form:

\[ a = 10^{3k} \]
\[ b = 10^{4k} \]
\[ c = 10^{5k} \]

### Step 3: Calculate \( c \times a \)

Next, we need to find \( c \times a \):

\[ c \times a = (10^{5k}) \times (10^{3k}) \]

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

\[ c \times a = 10^{5k + 3k} = 10^{8k} \]

### Step 4: Express \( c \times a \) in terms of \( b \)

We know from our earlier expression that:

\[ b = 10^{4k} \]

To express \( c \times a \) in terms of \( b \):

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

Thus, we have:

\[ c \times a = b^2 \]

### Conclusion

Therefore, the final result is:

\[ c \times a = b^2 \]