## Saturday, June 15, 2024

### Calculated Industries Qualifier Plus IIIFx: Logarithm, Exponential, Square Root

Calculated Industries Qualifier Plus IIIFx: Logarithm, Exponential, Square Root

Introduction

Say we have a financial calculator such as Calculated Industries Qualifier Plus IIIFx (or the Pocket Real Estate Master or another pocket basic financial calculator). Note that this calculator does not have the following functions: natural logarithm (ln x), exponential function (e^x), and square root (√). No problem. We can use the time value of money keys.

The relationship between present value (PV), future value (FV), periodic interest rate (I%), and number of periods (N) is determined as:

FV = PV * (1 + I% / 100) ^ N

For the natural logarithm and exponential, start by dividing both sides by PV:

FV / PV = (1 + I% / 100) ^ N

Let PV = 1 and equate 1 + I% / 100 = e (e ≈ 2.71828182846…)

Then:

1 + I% / 100 = 2.71828182846

I% / 100 = 1.71828182846

I% = 171.828182846

Since financial calculators are usually set to 2 decimal places, for most instances, we can use the approximation with reasonable accuracy:

I% ≈ 171.82818

Then we have:

FV / 1 = (1 + 171.82818 / 100) ^ N →

FV ≈ e^N and ln FV ≈ N

Now for square root:

FV / PV = (1 + I% / 100) ^ N

Let PV = 1, and N be 0.5 since √x is x^0.5 for x≥0.

Let x = 1 + I% / 100. Solving for I%:

x = 1 + I% / 100

x – 1 = I% / 100

100 * (x – 1) = I%

FV ≈ √x

FV / 1 = ((100 * (x – 1)) ^ 0.5

We can use this to build similar relationships with powers and roots.

Procedures and Examples

Procedure for ln(x):

Set the following variables:

P/Y = 1

Loan Amt = 1*

Int = 171.82818

FV = x

Solve for Term

Estimate ln 55.5.

P/Y = 1

Loan Amt = 1*

Int = 171.82818

FV = 55.5

Solve for Term

Result (Term): 4.02

Procedure for e^x:

P/Y = 1

Loan Amt = 1*

Int = 171.82818

Term = x

Solve for FV.

Estimate e^2.

P/Y = 1

Loan Amt = 1

Int = 171.82818

Term = 2

Solve for FV

Result (FV): 7.39

Procedure for √x:

P/Y = 1

Loan Amt = 1*

Term = 0.5

Int = 100 * (x – 1)

Solve for FV

Example: Estimate √84.

P/Y = 1

Loan Amt = 1

Term = 0.5

Int = 100 * (84 – 1) = 8300

Solve for FV

Result (FV): 9.17

*Note: Loan Amt (PV) is entered as -1 on graphing calculator TVM solvers.

Note: Please do not try this on the HP 12C or the BA Plus II, because on those calculators, the variable N is rounded up to the near highest integer.

Enjoy this hack and until next time,

Eddie

All original content copyright, © 2011-2024. Edward Shore. Unauthorized use and/or unauthorized distribution for commercial purposes without express and written permission from the author is strictly prohibited. This blog entry may be distributed for noncommercial purposes, provided that full credit is given to the author.

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