Point slop is one of the method used to find the straight line equation.
After understanding the exponential functionour next target is the natural logarithm. The natural log gives you the time needed to reach a certain level of growth. Read more about e. Natural Log ln is the amount of time needed to reach a certain level of continuous growth Not too bad, right?
E is About Growth The number e is about continuous growth. Natural Log is About Time The natural log is the inverse of e, a fancy term for opposite. Speaking of fancy, the Latin name is logarithmus naturali, giving the abbreviation ln.
Now what does this inverse or opposite stuff mean? After 3 units of time, we end up with If we want growth of The natural log gives us the time needed to hit our desired growth.
What is ln 1? Intuitively, the question is: How long do I wait to get 1x my current amount?
If we reverse it i. 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.
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. You can wiggle the variables all you want. The Rule of 72 The Rule of 72 is a mental math shortcut to estimate the time needed to double your money.
Now the question is easy: The Rule of 72 is useful for interest rates, population growth, bacteria cultures, and anything that grows exponentially. Where to from here? I hope the natural log makes more sense — it tells you the time needed for any amount of exponential growth.
The Natural Log of E Quick quiz: The math robot says:From March to September, the Dennis Rawlins page on Wikipedia was trashed repeatedly by the sort of dirty-fighter censors which establishments traditionally use to discourage exposure of what they're ever-hiding.
The slope of a line in the plane containing the x and y axes is generally represented by the letter m, and is defined as the change in the y coordinate divided by the corresponding change in the x coordinate, between two distinct points on the line.
This is described by the following equation: = . (The Greek letter delta, Δ, is commonly used in mathematics to mean "difference" or "change".). This is called the slope-intercept form because "m" is the slope and "b" gives the y-intercept.
(For a review of how this equation is used for graphing, look at slope and graphing.). I like slope-intercept form . Image Source: Google Images. Babies usually follow a straight line of increasing body length as they start growing.
This baby was born 20 inches long (y-intercept), and has been growing at .
Simply knowing how to take a linear equation and graph it is only half of the battle. You should also be able to come up with the equation if you're given the right information.
kcc1 Count to by ones and by tens. kcc2 Count forward beginning from a given number within the known sequence (instead of having to begin at 1).
kcc3 Write numbers from 0 to Represent a number of objects with a written numeral (with 0 representing a count of no objects). kcc4a When counting objects, say the number names in the standard order, pairing each object with one and only.