It’s time for logarithms• We’ll define their use

• We’ll show how to use them for different cases

• We’ll then go back to exponentials, and look at again how the spreadsheet work relates to the equations

And why repeatedly adding or subtracting a fraction is actually exponential; spreadsheets, apparently only adding or subtracting, and exponential eqns, do the same thing

• And THEN give you exponential and log equation to solve anything we will do in this course.

Now, logs• In exponentiation, if I take y=ax, then I am raising a to the

x power to get y(x)

• In taking the log, I am asking to what power did I raise a to get y? That is, I’m getting x

• EX:• What is y=23, or 2r. It’s y = 8;• Now ask, if I don’t know x, but I’m given 8, what is x?

• 8 = 2x, find x

Do this simple case, first• y = bx

• Take the log of both sides• log(y) = log(bx)

• TRUTH: log(bx) = x * log(b) AHA! x is OUTside the log

• log(y) = x * log(b). You are given y, and b a given constant

• log(y) / log(b) = x. Now look up/calculate log(y) and log(b)

Now with the starting point n.e. 1

• y = A * bx A and b are given constants• Take the log of both sides• log(y) = log(A * bx)• TRUTH: log( a * b ) = log( a ) + log( b )• TRUTH: log( bx ) = x * log( b ) AHA! x is OUTside the log

• log(y) = log(A) + x * log(b). You are given y, and b a given constant

• {log(y) / log(b)} – log(A) = x. Now look up/calculate log(y), log(b), and log(A)

Where do you get log calculations?• Computers, calculators, books. Calculate or look up.• Simple ones in your head. (Math secret #2)

• y(x) = (2)x

• Use logs to solve for exponent, use a number• When y is 8, what’s x?

• 8 = 2x

• log( 8 ) = log( 2x )• = x * log( 2 )• log(8)/log(2) = x;• do on calculator, or use Excel as calculator

Summary of log relationships• TRUTH: log(bx) = x * log(b)

• TRUTH: log( a * b ) = log( a ) + log( b )

• TRUTH: log( 1 ) = 0

• These are ALL you need for this course!

Slo-mo exponential, continued• DECREASING at a rate• Start with A• Remove (add) some fraction of A per unit of x• Get A – A * fraction, resulting in A * (1- fraction)

• Ex: reduce A by 3% per year, as in Russian population:• A(0) = A• A(1 year later) = A(1) = A(0) – ( A(0) * .03 )• (3% to a fraction is 3%/100 )

• THEN A(2 years later) = A(1) – { A(1) * .03 }• So to get next year, subtract 3% of this year.

Repeated removal (addition) of a fraction• . . . . Is equivalent to exponentiation, like this• Starting at A• A – A*(fraction) at end of first time interval• Simplify: A * (1-fraction) for decrease, • or A * (1+ fraction) for growth• Do it again• {A * (1- fraction) } * ( 1 – fraction ) = A * ( 1 – fraction ) 2

• If the (growth) is 100%, so fraction = 1, then y=A*(1+1)x

• should look like the Genie’s gift to you

Slo-mo spreadsheet and equation• They are the same. Repeatedly removing a fraction is the

same as exponentiation!

• In each cell of the spreadsheet, we do this:

• A - A*(fraction) = A * (1 - fraction)• at end of time interval

• And doing that over and over results in this:

• A * (1 – fraction) x or this A * (1+ fraction)x

• Where x is the number of intervals on x (often time)

The special “leaky bucket”• It always leaks a fraction of what’s left

So this is exponential by definition

• Because we are changing a value by• A constant fraction = rate = percentage, all meaning the

same

• Per interval of x (which is often time)

FINALLY, “fast motion”

How many pennies does the Genie give on the 6th day?

y(days) = .01 * ( 1 + 1)days

• = .01 * ( 2 )5 = 0.32

How many days does it take to get up to 32 cents?

0.32 = .01 * ( 1 + 1 )x

log(0.32) = log(0.01) + log(2x)

= log(0.01) + x * log(2) it’s algebra now!

{ log(0.32) – log(0.01) } / log(2) = 5

Here’s what you must know• TRUTH: log(bx) = x * log(b)• TRUTH: log( a * b ) = log( a ) + log( b )• TRUTH: log( 1 ) = 0

• y(x) = A * (1 + rate)x

• Or• y(x) = A * (1 – rate)x

• Where A is the starting point, and x is the number of intervals of the independent, or controlling, variable.