How do I use logs to find the value of these expressions?

Question

What is the value of these expressions? I'm supposed to leave them in three significant digits.

1. [(92.5)(.345)] / 17.4

2. I don't know how to put this. There is a "4" in superscript, followd by the root symbol, with the number .00439 in it? The fourth root of .00439?

3. [(.391)(97.2)]^2

Answer 1

Few simple facts:

log(a * b) = log(a) + log(b)
log(a/b) = log(a) - log(b)
log(a^b) = b * log(a)

1. [(92.5)(.345)] / 17.4
so we need log(92.5) + log(.345) - log(17.4) =
1.96614 + -0.46218 - 1.24055 = 0.26341
We now need 10^0.26341 = 1.834

2: .00439^(1/4)
log(.00439^(1/4)) = 1/4 * log( .00439) = 1/4 * -2.35754
= -.58939

We now need 10^-.58939 = 0.257

3: [(.391)(97.2)]^2
log([(.391)(97.2)]^2) = 2*log([(.391)(97.2)]) =
2*(log(.391)+ log(97.2)) =
2*( -0.40782 + 1.98767) =
2 * 1.57985 =
3.1597

We now need 10^3.1597 = 1444.442

Answer 2

If you're supposed to use logarithms to find the answers, then you will need to use these identities (facts) about logarithms...

log (ab) = log a + log b
(for example log 6 = log 2 + log 3)

log (a/b) = log a - log b
(for example log 6 = log 18 - log 3)

log (a^b) = b x log a
(for example log 9 = 2 x log 3)

For question 2, you are correct - the symbol you describe does mean "the fourth root". This is the same as raising to the power 1/4, which lets you use the last identity I have given above.

I hope this helps.

Answer 3

1. set it up like this.

[log92.5 + log.345] - log17.4 =

1.966 + (-.4621) - 1.2405 = 0.2634

antilog 0.2634= 1.83405

2. sounds like 4th root of 0.00439 to me

3. set it up like this.

log X^2 = 2*log X

[ log 0.391 + log 97.2]^2 = 2 X [ log 0.391 + log 97.2]

2*(-.40782 + 1.98767) = 2* (1.57984) = 3.159686

antilog 3.159686 = 1444.3952

hope that helps. logs can be a pain if you are not used to working them.