Approximate the logarithm using the properties of logarithms, given logb 2 ≈ 0.3562, logb 3 ≈ 0.5646, and logb 5 ≈ 0.8271. (Round your answer to four decimal places.)

Question

Approximate the logarithm using the properties of logarithms, given log_b 2 ≈ 0.3562,  log_b 3 ≈ 0.5646,

 and log_b 5 ≈ 0.8271.  (Round your answer to four decimal places.)

log_{b^2 3b}

Answer 1

Hello Allen,

Since all the information you are given relates to the log base b, but you want to calculate a logarithm relative to the base b², you will need to use the change of base formula. In general, this can be written as

log_d(x) = log_c(x)/log_c(d).

In this case, we can write

log_b^2(x) = log_b(x)/log_b(b²)

But since log_b(b²)= 2, we have

(Eq. 1) log_b^2(x) = log_b(x)/2

Now we can calculate log_b^2(3b).

log_b^2(3b) = log_b^2(3) + log_b^2(b)

log_b^2(3b) = [log_b(3)]/2 + [log_b(b)]/2 (using Eq. 1.)

log_b^2(3b) ≈ .5646/2 + 1/2

log_b^2(3b) ≈ .2823 + .5000

log_b^2(3b) ≈ .7823 (to four decimal places.)

Hope that helps! Let me know if you need any further explanation.

William