# 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 ruleNumerical 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)]