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Understanding Logarithms and Their Relationship with Exponentials in Excel

This course module explores logarithms and their applications, demonstrating how to utilize them alongside exponential equations in various scenarios. We will define logarithms and relate them to spreadsheets, highlighting the connection between addition/subtraction of fractions and exponential functions. Through practical examples, you'll learn how to solve equations using logarithms, handle exponential growth and decay, and see how spreadsheets replicate mathematical behaviors. By the end, you'll master the concepts of logs and exponentials essential for course success.

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Understanding Logarithms and Their Relationship with Exponentials in Excel

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Presentation Transcript

  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.

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