8log8(5) − log8(7) = ?
log8 (5^8) - log8(7)
= log8 (5^8/7)
To solve this problem, we can use the rules of logarithms. In particular, we can use the product rule of logarithms, which states that log base b of x minus log base b of y is equal to log base b of (x/y).
In this case, we can rewrite the expression using the product rule:
8log8(5) − log8(7) = log8(5/7)
Now, let's simplify this further:
log8(5/7)
To evaluate this logarithm, we need to determine what power of 8 gives us the value of 5/7. In other words, we need to find x such that 8^x = 5/7.
To do this, we can rewrite 5/7 as a power of 8:
5/7 = (8^(1/3))^x
Now, we equate the exponents:
8^x = (8^(1/3))^x
Since the bases are the same, the exponents must also be the same:
x = 1/3
So, the answer is:
8log8(5) − log8(7) = log8(5/7) = 1/3