How can you express 6log_2(m) + log_2(x)/3 as a single logarithm?

To combine the expression ( 6\log_2(m) + \frac{\log_2(x)}{3} ) into a single logarithm, we can use the properties of logarithms.

The property ( a\log_b(c) = \log_b(c^a) ) allows us to move coefficients in front of logs into the exponent of their arguments. This means:

1. The term ( 6\log_2(m) ) can be expressed as ( \log_2(m^6) ).

2. The term ( \frac{\log_2(x)}{3} ) can be rewritten as ( \log_2(x^{1/3}) ) because ( \frac{1}{3} ) is the same as raising ( x ) to the power of ( \frac{1}{3} ).

Now, we have converted the original expression into:

[

\log_2(m^6) + \log_2(x^{1/3})

]

Using the logarithmic property ( \log_b(a) + \log_b(c) = \log_b(a \cdot c) ), we can combine these two logarithms: