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.





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