It ’ s time for logarithms

1. 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.

2. 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

3. 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)

4. Now with the starting point n.e. 1 • y = A * bxA 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)

5. 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

6. 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!

7. 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.

8. 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

9. 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) xor this A * (1+ fraction)x • Where x is the number of intervals on x (often time)

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

11. 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)

12. 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

13. 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.