Help with Logarithms?

Question

Can anyone show me how to do the following logarithm without a calculator.

N= the cubed root of [ (9.1)^2 + (3.52) ]

I'm supposed to solve it by using logarithms (taking the log of each side)

Answer 1

Well, first we must get rid of the cubic root by cubing both sides of the equation:

N^3 = 9.1^2 + 3.52

Now we take the logarithm base N of both sides:

log(N)N^3 = log(N)[9.1^2 + 3.52]

3 = log(N)[9.1^2 + 3.52]

Now we use the change of base formula using base 10:

3 = log(9.1^2 + 3.52)/log(N)

Now we expand using the quotient rule:

3 = log(9.1^2 + 3.52) - log(N)

Now we use paper and pencil to find 9.1^2

It turns out to be:

82.81

Now we add 3.52 to it:

86.33

Ok, now we resimplify it:

3 = log(86.33) - log(N)

Crunch it again using the quotient rule:

3 = log(86.33)/log(N)

move the log(N) to the other side:

3log(N) = log(86.33)

Now we move the three to the other side:

log(N) = log(86.33)/3

N = 10^(log(86.33)/3)

And you're done.

Answer 2

You can simply the expression inside the radical using multiplication and addition, but you can't get a good estimate of that cube root without either calculus or a calculator. Also I'm not sure where you're seeing the logarithm, because there aren't any in this problem.

Answer 3

If you have a newer Texas Instrument: Go to MATH, down to number 4, hit enter, and you can try parenthesis instead of brackets cause I don't think they have them on calculator. Then when you type it in that should be the answer, but no promises. I'm not a math whiz.=]

Answer 4

N = [ 9.1² + 3.52 ]^(1/3)
N = 86.33^(1/3)
ln N = (1/3).ln 86.33
ln N = (1/3) x 4.458
ln N = 1.486
N = e^(1.486)
N = 4.42