RPN: HP 11C: Transferring Between Bases (Common/Natural)

RPN: HP 11C: Transferring Between Bases (Common/Natural)

All algorithms were tested with the HP 11C.

Between the Exponential Function and Common-Antilog

e^α = 10^ß

Given α, what is ß?

ß = log(e^α)

RPN:

<input α>

e^x

LOG

Examples (Fix 6):

e^1.05 = 10^ß

ß ≈ 0.456009

e^(-2.2) = 10^ß

ß ≈ -0.955448

Given ß, what is α?

α = ln(10^ß)

RPN:

<input ß>

10^x

LN

Examples (Fix 6):

e^α = 10^5.4

α ≈ 12.433960

e^α = 10^0.366

α ≈ 0.842746

Between the Natural Logarithm and Common Logarithm

log α = ln ß

Given α, what is ß?

ẞ = exp(log α)

RPN:

<input α>

LOG

e^x

Examples (Fix 6):

log 17 = ln ß

ß ≈ 3.422766

log 317 = ln ß

ß ≈ 12.195405

Given ß, what is α?

α = 10^(ln ß)

RPN:

<input ß>

LN

10^x

Examples (Fix 6):

log α = ln 425

α ≈ 1,127,428.915

log α = ln 9.81

α ≈ 192.044677

Eddie

All original content copyright, © 2011-2026. 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.

Approximations for Common Logarithm Function

Approximations for Common Logarithm Function

A Personal Note:

Hi, everyone!  I had heart surgery two weeks ago, and I am still in recovery.  However, things are going well and I am breathing a lot easier.  Glad to be back. 
- Eddie

Transformations

I am trying to find an approximation polynomial for the common logarithmic function, log(x).   My goal is to find an approximation polynomial which is accurate to at least 2 decimal points.

The first fit was to fit x against log(x).  However, if I apply a transformation, then compare data, I was able to get better results.

I used a TI-84 Plus CE to fit a quadratic and cubic polynomial to data generated by the following sets:

x,  log(x)
x,  log(x^(1/2))
x,  log(x^(1/3))
x,  log(x^(1/4))

Comparison of Approximations

The table compares two approximations against the logarithmic function.

POLY 1:
log x ≈ -.63965*t^2 + 3.06651*t - 2.44635

POLY 2:
log x ≈ .39510*t^3 - 2.10974*t^2 + 4.805*t - 3.091

In both polynomials, t = x^(1/4)

Eddie

All original content copyright, © 2011-2020.  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.