Given that :

log a = 12

log b= 3

We need to find the value of the expression :

log (a^2*b^3)^6

First we know that log a^b= b*log a.

==> log (a^2*b^3)^6 = 6*log (a^2*b^3)

Now we know that log ab = log a + log b

==> log (a^2*b^3)^6 = 6[ log a^2 + log b^3]

= 6 [ 2log a + 3log b}

Now we simplified as follows.

==> log (a^2*b^3)^6 = 6[ 2log a + 3log b]

We will substitute with the given values.

==> log (a^2*b^3)^6 = 6( 2*12 + 3*3)

= 6(24+9)

= 6*33

= 198

**Then the value of log (a^2*b^3)^6 = 198.**

It is given that log a = 12 and log b = 3

To determine log (a^2* b^3)^6

use the property of logarithms : log a*b = log + log b and log a^b = b*log a

log (a^2* b^3)^6

=> 6*log (a^2* b^3)

=> 6*log a^2 + 6*log b^3

=> 12*log a + 18*log b

=> 12*12 + 18*3

=> 144 + 54

=> 198

**The value of log (a^2* b^3)^6 = 198**

To find the value of `log(a^2* b^3)^6` use the following properties of logarithm.

log a^b = b*log a, and log a*b = log a + log b.

Using these: log(a^2* b^3)^6

= 6*log (a^2* b^3)

= 6*(log a^2 + log b^3)

= 6*(2*log a + 3*log b)

= 12*log a + 18*log b

Now, log a = 12 and log b = 3

Substituting these values gives 12*12 + 18*3 = 144 + 54 = 198

The value of log(a^2* b^3)^6 is 198