Let log_b 2 = A and log_b 3 = C. Write the expression below in terms of A and C.

Question

log_{b sqrt(128/3)}

Answer 1

Hi Alexandria,

 

I am assuming the question is asking for log_b(sqrt(128/3))

 

sqrt(128/3) = √128 / √3

logb(sqrt(128/3)) = logb(√128 / √3)

 

Based on the quotient identity property of logs, we can write

logb(√128 / √3) = logb(√128) - logb(√3)

logb(√128) = logb(√64*2) 

logb(√64*2) = logb(8*√2)

based on the product identity of logs, we can write

logb(8*√2) = logb(8) + logb(√2)

logb(8) + logb(√2) = logb(2^3) + logb(2^1/2)

logb(2^3) + logb(3^1/2) = 3*logb(2) + 1/2*logb(2)

3*logb(2) + 1/2*logb(2) = 3A + (1/2)A

logb(√128) = 3A + (1/2)A = (7/2)A

now on to logb(√3)

logb(√3) = logb(3^1/2)

= 1/2*logb(3)

= 1/2C

 

so 

logb(sqrt(128/3)) = (7/2)A - (1/2)C