# caluclate log8 (4).

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We have to find the value of log(8)4.

Let log(8) 4 = x

=> 4 = 8^x

=> 2^2 = 2^3^x

=> 2^2 = 2^3x

As the base is the same, we equate the exponent

=> 3x = 2

=> x = 2/3

**The required value is log(8) 4 = 2/3**

We need to find the values of log8 (4)

We will use logarithm properties to solve.

We know that:

loga b = logc b/ logc a

Then we will rewrite:

log8 4 = log2 4 / log2 8

Now we will simplify:

==> log8 4 = log2 2^2 / log2 2^3

==> log8 4 = 2log2 2 / 3log2 2

But log2 2 = 1

**==> log8 4 = 2/3 **

We'll interchange the base and argument and we'll get:

log8 (4) = 1/log4 (8)

But 8 = 4*2

log8 (4) = 1/log4 (4*2)

We'll use the product property of logarithms:

log4 (4*2) = log4 (4) + log4 (2) = 1 + log4 (2)

But log4 (2) = 1/log2 (4) = 1/log2 (2^2)

We'll use the power property of logarithms and log2 (2) = 1

1/log2 (2^2) = 1/2log2 (2) = 1/2

1/log4 (4*2) = 1/[1 + log4 (2)] = 1/(1 + 1/2) = 1/[(2+1)/2] = 1/(3/2) = 2/3

**Therefore, the requested value for log8 (4) = 2/3.**