Rules of logarithms

Quick guide

This is a summary of the rules of logarithms material used in the xlf presentation series. It include numerical examples, and keystrokes for the Hewlett Packard 12C financial calculator.

$log(x)$ normally represents the logarithm with base $e$
e's value is $2.718281828459045$ to 16 digits, or $2.71828182845904$ at the Excel maximum of 15 digits
Using the binomial expansion $e$ is calculated as $$e=\lim_{n\rightarrow\infty}\left(1+\frac{1}{n} \right)^n \ $$ or $$e=\sum_{n=0}^{\infty}\left(\frac{1}{n!} \right) \ $$

RPN keystrokes ►

Logarithms

In the expression $x^j$, $j$ is the exponent and $x$ is the base.

Log rule	Numerical example
$x^0=1$	$5^0=1$ $\text{12c: }5\;[\text{Enter}]\;0\;[y^x]\;\text{returns} \;1$
$x^1=x$	$4^1=4$ $\text{12c: }4\;[\text{Enter}]\;1\;[y^x]\;\text{returns} \;4$
$x^{-1}=1/x$	$2^{-1}=1/2$ $\text{12c: }2\;[\text{Enter}]\;1\;[\text{CHS}]\;[y^x]\;\text{returns} \;0.5$
Product rule $x^j x^k=x^{j+k}$	$x^2x^4=x^{2+4}=x^6$ $2^2\cdot2^4=4\cdot16=64; \quad 2^6=64$ $\text{12c: }2\;[\text{Enter}]\;2\;[y^x]\;2\;[\text{Enter}]\;4\;[y^x]\;[×]\;\text{returns}\;64$
Quotient rule $x^j/x^k=x^{j-k}$	$x^7/x^3=x^{7-3}=x^4$ $5^7/5^3=5^{7-3}=5^4=625; \quad 78125/125=625$ $\text{12c: }5\;[\text{Enter}]\;7\;[y^x]\;5\;[\text{Enter}]\;3\;[y^x]\;[÷]\;\text{returns} \;625$
Power rule $(x^j)^k=x^{jk}$	$(x^2)^4=x^{2\cdot4}=x^8$ $(4^2)^4=4^8=65536; \quad 16^4=65536$ $\text{12c: }4\;[\text{Enter}]\;2\;[y^x]\;\;4\;[y^x]\;\text{returns} \;65536$
$(xy)^j=x^jy^j$	$(xy)^3=x^3y^3$ $(2\cdot3)^3=2^3\cdot3^3=8\cdot27=216; \quad 6^3=216$ $\text{12c: }2\;[\text{Enter}]\;3\;[×]\;3\;[y^x]\;\text{returns}\;216$
$(x/y)^j=x^j/y^j$	$(x/y)^2=x^2/y^2$ $(4/5)^2=4^2/5^2=16/25=0.64; \quad(0.8)^2=0.64$ $\text{12c: }4\;[\text{Enter}]\;5\;[÷]\;2\;[y^x]\;\text{returns}\;0.64$
$x^{-j}=1/x^j$	$x^{-3}=1/x^3$ $2^{-3}=1/2^3=1/8; \quad2^{-3}=0.125$ $\text{12c: }2\;[\text{Enter}]\;3\;[\text{CHS}]\;[y^x]\;\text{returns}\;0.125$

Note 1: $\dfrac{x^j} {x^j}=1$, thus by application of the quotient rule $x^{j-j}=x^0=1$

References

Imperial College London, Euler's Identity' Divide and Rule, Stories from the Complex Plane (page 2), accessed 30 January 2020

Published: 2 February 2015
Revised: Thursday 30th of January 2020 - 04:18 PM, [Australian Eastern Standard Time (EST)]

Log rule	Numerical example
\(x^0=1\)	\(5^0=1\) \(\text{12c: }5\;[\text{Enter}]\;0\;[y^x]\;\text{returns} \;1\)
\(x^1=x\)	\(4^1=4\) \(\text{12c: }4\;[\text{Enter}]\;1\;[y^x]\;\text{returns} \;4\)
\(x^{-1}=1/x\)	\(2^{-1}=1/2\) \(\text{12c: }2\;[\text{Enter}]\;1\;[\text{CHS}]\;[y^x]\;\text{returns} \;0.5\)
Product rule \(x^j x^k=x^{j+k}\)	\(x^2x^4=x^{2+4}=x^6\) \(2^2\cdot2^4=4\cdot16=64; \quad 2^6=64\) \(\text{12c: }2\;[\text{Enter}]\;2\;[y^x]\;2\;[\text{Enter}]\;4\;[y^x]\;[×]\;\text{returns}\;64\)
Quotient rule \(x^j/x^k=x^{j-k}\)	\(x^7/x^3=x^{7-3}=x^4\) \(5^7/5^3=5^{7-3}=5^4=625; \quad 78125/125=625\) \(\text{12c: }5\;[\text{Enter}]\;7\;[y^x]\;5\;[\text{Enter}]\;3\;[y^x]\;[÷]\;\text{returns} \;625\)
Power rule \((x^j)^k=x^{jk}\)	\((x^2)^4=x^{2\cdot4}=x^8\) \((4^2)^4=4^8=65536; \quad 16^4=65536\) \(\text{12c: }4\;[\text{Enter}]\;2\;[y^x]\;\;4\;[y^x]\;\text{returns} \;65536\)
\((xy)^j=x^jy^j\)	\((xy)^3=x^3y^3\) \((2\cdot3)^3=2^3\cdot3^3=8\cdot27=216; \quad 6^3=216\) \(\text{12c: }2\;[\text{Enter}]\;3\;[×]\;3\;[y^x]\;\text{returns}\;216\)
\((x/y)^j=x^j/y^j\)	\((x/y)^2=x^2/y^2\) \((4/5)^2=4^2/5^2=16/25=0.64; \quad(0.8)^2=0.64\) \(\text{12c: }4\;[\text{Enter}]\;5\;[÷]\;2\;[y^x]\;\text{returns}\;0.64\)
\(x^{-j}=1/x^j\)	\(x^{-3}=1/x^3\) \(2^{-3}=1/2^3=1/8; \quad2^{-3}=0.125\) \(\text{12c: }2\;[\text{Enter}]\;3\;[\text{CHS}]\;[y^x]\;\text{returns}\;0.125\)