Andre says that log 55 =1.5 because 55 is halfway between 10 and 100.

Question

Andre says that log 55 =1.5 because 55 is halfway between 10 and 100. Do you agree. Explain your reasoning.

(Question is base-10 logarithm)

Answer 1

the thing is Andre is claiming log₁₀(55)=1.5

which is wrong

and claiming the reason is f(x)=log₁₀(x) is linear

which is wrong

the linear function 1/90=(y-1)/(x-10)

y=(x/90)+80/90

has solution x=55, y=1.5

log₁₀(55) is not 1.5

the only two points where the linear equation is accurate

are at x=1 and x=2, the precise points picked

to create the line.

y=log₁₀(x) is a curve not a line.

at x=o the curve goes toward negative infinity

to find log(x/2) use log(x/2)=logx-log2, and learn what log₁₀(2) is. for log base 10, log₁₀(2)=.3,,,for natural base e, ln2=.69.,,,,so log₁₀(50)=2-.3=1.7,,,,,if I know ln(20.1)=3 then ln(10.05)=3-.69=2.31

Answer 2

Hi Lorena,

Depends on how precise you want to get. log 55 ≈ 1.74, however, 1.5 could be close enough of a guess for certain purposes.

This is because the log function is not perfectly linear, though a linear approximation can be good enough at times. Since it is not perfectly linear, it is not entirely correct to just take the half point between two known values. What Andrew did is the linear approximation, but it could be good enough for most purposes.

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