given that c is the logarithm of x in base a, and d is the logarithm of x in base b, find the general statement that expresses the log of x in base ab, in terms of c and d.
[*subscripts not included: a,b,ab]
loga_x=c ; logb_x=d
...logab_x = [in terms of c and d]
To find the general statement that expresses the logarithm of x in base ab, in terms of c and d, we can make use of the change of base formula.
The change of base formula states that for any positive numbers x, a, and b, and any positive integers m and n:
loga(x) = logb(x) / logb(a)
Using this formula, we can express the logarithm of x in base ab, in terms of c and d as follows:
logab(x) = logb(x) / logb(ab)
Now, we know that the logarithm of x in base a is given by:
loga(x) = c
Similarly, the logarithm of x in base b is given by:
logb(x) = d
Substituting these values into the equation:
logab(x) = logb(x) / logb(ab)
logab(x) = d / logb(ab)
Thus, the general statement that expresses the logarithm of x in base ab, in terms of c and d, is:
logab(x) = d / logb(ab)