approximate the logarithm

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 20

Answer 1

b = 6.9994

logb(20)= 1.5396

b^.5646 = 3

b = the .5646 root of 3 = 6.99963

b^x = 20

6.99963^x = 10

6.99963 is the xth root of 10

the 1.5396th root of 10 = 6.99963

It's between 10 and the sqr10 or 3.2 < x < 10 the mid point is close to 7

Answer 2

log_b20 = log_b(2²•5) = log_b(2²) + log_b5 = 2log_b2 + log_b5 ≈ 2(0.3562) + 0.8271 = 1.5395