If log(a) 2 + log(a) 4 = log (2) a^2, what is a?

We have log(a) 2 + log(a) 4 = log (2) a^2

convert all the terms to a form with logarithm to the base 2

=> log 2 / log a + log 4 / log a = log a^2 / log 2

=> (1/ log a)( log 2 + log 4) = 2*log a / log 2

Use log a + log b = log a*b and log 2 = 1

=> log 8 / log a = 2* log a

=> log 8 = 2* (log a)^2

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

=> (log a)^2 = (3/2)

=> log a = sqrt (3/2)

a = 2^(sqrt (3/2))

a = 2^(sqrt 1.5)

**The required value of a is 2^(sqrt 1.5)**

log(a) 2 + log(a) 4 = log(2) a^2 .

We will use logarithm properties to find a.

We know that log a + log b = log a*b

==> log(a) 2 + log(a) 4 = log(a) 2*4 = log(a) 8

==> log(a) 8 = log(2) a^2

Now we will rewrite:

log(a) 8 = log(2) 8 / log(2) a = log(2) 2^3 / log(2) a = 3/log(2) a

==> 3/log(2) a = log(2) a^2

==> 3 = 2log(a) 2 * log(2) a

==> 2[log(2) a)'^2 = 3

==> log(2) a = sqrt(3/2)= sqrt(1.5)

==> Now we will rewrite into exponent form.

**==> a = 2^(sqrt(1.5)**