Laws of Logarithms


 Laws of Logarithms

Equation in the form of index and logarithm:

\(N=a^x \iff \log_aN=x\)


where \(a \gt0,\, a \ne1\).


\(\begin{aligned} &\bullet \log_aa^x=x \\\\ &\bullet \log_a1=0 \\\\ &\bullet \log_aa=1 \end{aligned}\)

The diagram shows the graphs of exponential and logarithmic functions.


We can see that the exponential and logarithmic functions are reflection of one another in the straight line \(y=x\).

The exponential and logarithmic functions are inverse functions of one another.

The logarithms of negative numbers and of zero are undefined.
Law of logarithms:

\(\begin{aligned} &\bullet \log_axy=\log_ax+\log_ay \\\\ &\bullet \log_a\dfrac{x}{y}=\log_ax-\log_ay\\\\ &\bullet \log_ax^n=n \log_ax \end{aligned}\)


for any real number \(n\)

where \(a,x\) and \(y\) are positive numbers and \(a\ne1\).

Change of base of logarithms:

\(\begin{aligned} &\bullet \log_ab=\dfrac{\log_cb}{\log_ca} \\\\ &\bullet \log_ab=\dfrac{1}{\log_ba} \end{aligned}\)

where \(a\), \(b\) and \(c\) are positive numbers, \(a \ne 1\) and \(c \ne 1\).
\(\lg=\log_{10}\) (common logarithms) and \(\ln=\log_{e}\) (natural logarithms) where \(e\) is a constant.