Which value corresponds to the logarithm log base 8 of 2?

To understand why the value corresponds to logarithm base 8 of 2 is ( \frac{1}{3} ), we start with the definition of logarithms. The logarithm ( \log_b(a) = c ) means that ( b^c = a ). In this case, we want to find the value of ( c ) such that:

[

8^c = 2

]

We know that ( 8 ) can be expressed as ( 2^3 ), so we can rewrite the equation:

[

(2^3)^c = 2

]

Using the power of a power property in exponents, we can simplify the left side:

[

2^{3c} = 2^1

]

Since the bases are the same, we can set the exponents equal to each other:

[

3c = 1

]

Now, we solve for ( c ):

[

c = \frac{1}{3}

]

Thus, ( \log_8(2) = \frac{1}{3} ). This confirms that the correct value for ( \log_8(2) ) is indeed (