If a = log5\sqrt{13 } and 13^2b=5, then ab =

Question

a base is 5 and it has a square root of 13

Answer 1

In other words 5^x=13^.5

" =3.6055....

x=.796

Answer 2

If you convert the equation including b into a logarithmic equation, you could write that log₁₃5=2b.

If you solve for b, you now have 0.5log₁₃5=b.

Remember that sqrt(13) = 13^0.5, so log₅sqrt(13) = log₅13^0.5. When we have a power inside a log, we can move it to being a coefficient in the front. Thus, log₅sqrt(13) = log₅13^0.5 = 0.5log₅13 = a.

Thus, ab = (0.5log₅(13))*(0.5log₁₃5)

Let's use the change of base formula to make this easier to deal with. Recall that the change of base formula says that log_ba = ln(a)/ln(b). (If you don't want to use ln, you can use any other base, not just base e.)

So, a = 0.5log₅(13) = 0.5ln(13)/ln(5), and b = 0.5log₁₃5 = 0.5ln(5)/ln(13)

Now, ab = (0.5ln(13)/ln(5))*(0.5ln(5)/ln(13)) = 0.5*0.5 (the ln(5) and ln(13) both cancel).

Thus, ab = 0.5*0.5 = 0.25