The Common and Natural Logarithms?

Question

1. AntiLn(3)

2. Ln -e^2

3. Antilog(-2.7)

4. Ln (-e)^2

5. Antilog 3.296

Answer 1

1. 3
2. undefined
3. exp(-2.7)
4. 2

Answer 2

The general logarithm is f(x)=K*log(x), where K is a constant. The only differece between common and natural logarithms is the factor K. Natural logarithms use a K of (1/log(e)) where e=about 2.718281828. Common logarithms use a K of (1/log(10)). This formula works no matter which of the log functions you start with. I prefer log(x) as the natural logarithm where log(e)=1. Then common logarithms are then f(x)=log(x)/log(10). Now to your questions:

1: AntiLn(3) = exp(3) = 2.71828...^3 = e*e*e
= 20.085536923187667740928529654582

2: log is not defined for negative numbers

3: Antilog(-2.7) = 10^-2.7 = 1/ (10^2.7) = 1/exp(log(10^2.7)) = 1/exp(2.7*log(10)/log(e)) = 1/exp(2.3025850929940*2.7)
= 1/exp(6.21697975108392334)
= 1/501.187233627272285
= 0.0019952623149688796013524553967395

4: log(-e)^2/log(e) = log(e^2)/log(e) = 2*log(e)/log(e) = 2

5: antilog 3.296 = 10^3.296 = exp(log(10^3.296)/log(e)) = exp(3.296*log(10)/log(e)) = exp(3.296*2.3025850929940)
= exp(7.5893204665083745745232)
= 1976.96964011186077215302090

By the way, log(10)/log(e) comes up often enough in numerical computation that late moel Intel processors have an instruction that loads the constant to a floating-point register.

Answer 3

1. AntiLn(3) = Exp(3) = 20.0855369 (just google: exp(3) and you'll have the answer).

2. Ln (-e^2) = Ln(e^2) + Ln(-1) = 2 + Pi * I
(Pi = 3.1415..., I: Imaginary unit);

3. Antilog(-2.7) = 10^(-2.7) = 0.00199526231

4. Ln(-e)^2 = Ln (e^2) = 2
(Did you mean: (Ln(-e))^2?:
(Ln(-e))^2 = (Ln(-1) + Ln(e))^2 = (1+Pi*I)^2

5. Antilog(3.296) = 10^3.296 = 1 976.96964

Answer 4

Use your calculator.

This is how I do it on a TI-83 or TI-84, etc graphing calculator

1. antiln(3)
Press "2nd" and then "ln" to get e^(
press "3" and ")"
press "="

2. ln -e^2
press "ln", negative sign "-" not minus sign, "2nd", "ln", "2"

3. antilog(-2.7)
type in "10^-2.7"

4. ln (-e)^2
press "ln", negative sign "-", "2nd", "ln", "2"

5. antilog 3.296
type in "10^-3.296"

Antilog of x means 10^x

And of course, you press "=" of each of them to get the answer =)

Answer 5

1. 20.08553692

2. 2 + pi i

3. .0019952623

4. -8.869604401+2pi i

5. 1976.96964

note: by i I mean sqare root -1

Answer 6

20 .0855

no real answer

.00199526

2

1976.9696