If log 10 to base 8 = X, evaluate log 5 to base 8 in terms of X.
Answer Details
We can use the change of base formula to evaluate log 5 to base 8 in terms of X:
log 5 to base 8 = log 5 / log 8
We know that log 10 to base 8 = X, which means:
8^X = 10
We can rewrite this as:
2^3X = 10
Taking the logarithm of both sides of the equation with base 2, we get:
log 2 (2^3X) = log 2 10
3X = log 2 10
X = (1/3) log 2 10
Substituting this value of X into the expression for log 5 to base 8, we get:
log 5 to base 8 = log 5 / log 8
= (log 5 / log 2) / (log 8 / log 2)
= log 2 5 / log 2 8
= log 2 5 / (3 log 2 2)
= (1/3) log 2 5
Therefore, log 5 to base 8 in terms of X is:
X - (1/3) log 2 5.