# Natural log and relationship message

### Intro to Logarithms (article) | Khan Academy

In mathematics, the logarithm is the inverse function to exponentiation. That means the logarithm of a given number x is the exponent to which The natural logarithm has the number e (that is b ≈ ) as its base; its use is the amount of information conveyed by any one such message is quantified as log2( N) bits. Natural Log (ln) is the amount of time needed to reach a certain level of continuous This relationship makes sense when you think in terms of time to grow. base 10 logarithm (log), natural logarithm (ln), exponential function (exp or ex) This relationship is described by the equation. y = 10 x,. and described by this.

Why is a logarithm useful?

### Demystifying the Natural Logarithm (ln) – BetterExplained

And you'll see that it has very interesting properties later on. But you didn't necessarily have to use algebra. To do it this way, to say that 'x' is the power you raise 3 to to get to 81, you had to use algebra here, while with just a straight up logarithmic expression, you didn't really have to use any algebra, we didn't have to say that it was equal to 'x', we could just say that this evaluates to the power I need to raise 3 to to get to The power I need to raise 3 to to get to Well what power do you have to raise 3 to to get to 81?

Well let's experiment a little bit, so 3 to the first power is just 3, 3 to the second power is 9, 3 to the third power is 27, 3 to the fourth power, 27 times 3 is equal to So we could say Log, base 3, of 81, is equal to-- I'll do this in a different colour. Is equal to 4. Let's do several more of these examples and I really encourage you to give a shot on your own and hopefully you'll get the hang of it.

### Intro to logarithms (video) | Khan Academy

So let's try a larger number, let's say we want to take log, base 6, of What will this evaluate to? Well we're asking ourselves, "what power do we have to raise 6 to, to get ? This is equal to So this is 6 to the third power is equal to So if someone says "what power do I have to raise 6 to-- this base here, to get to ?

Let's try another one.

## Intro to logarithms

Let's say I had, I dunno, log, base 2, of So what does this evaluate to? Well once again we're asking ourselves, "well this will evaluate to the exponent that I have to raise this base 2, and you do this as a little subscript right here. The exponent that I have to raise 2 to, to get to So this right over here is 2 to the sixth power, is equal to So when you evaluate this expression you say "what power do I have to raise 2 to, to get to 64? Let's do a slightly more straightforward one, or maybe this will be less straightforward depending on how you view it.

What is log, baseof 1?

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I'll let you think about that for a second. How long do I wait to get 1x my current amount? If we reverse it i.

**Common and Natural Logarithms**

If we go backwards. In general, you can flip the fraction and take the negative: This means if we go back 1. Ok, how about the natural log of a negative number?

Logarithmic Multiplication is Mighty Fun How long does it take to grow 4x your current amount? Sure, we could just use ln 4. We can consider 4x growth as doubling taking ln 2 units of time and then doubling again taking another ln 2 units of time: Any growth number, like 20, can be considered 2x growth followed by 10x growth.

## Demystifying the Natural Logarithm (ln)

Or 4x growth followed by 5x growth. Or 3x growth followed by 6. This relationship makes sense when you think in terms of time to grow. If we want to grow 30x, we can wait ln 30 all at once, or simply wait ln 3to triple, then wait ln 10to grow 10x again.

The net effect is the same, so the net time should be the same too and it is.

Well, growing 5 times is ln 5. Suppose we want 30x growth: We can consider the equation to be: If I double the rate of growth, I halve the time needed. The natural log can be used with any interest rate or time as long as their product is the same.