JEE Mathematics Logarithms Notes Set 03

Definition and Basic Properties

For \( a>0,\ a\ne1 \) and \( N>0 \), \( \log_aN=x \) means \( a^x=N \). Hence \( a^{\log_aN}=N \), \( \log_a1=0 \) and \( \log_aa=1 \).
Product, quotient and power rules: \( \log_a(MN)=\log_aM+\log_aN \), \( \log_a\dfrac MN=\log_aM-\log_aN \), \( \log_aM^k=k\log_aM \) for positive M, N.

Change of Base

\( \log_aN=\dfrac{\log_bN}{\log_ba} \) for any valid base b. In particular \( \log_ab=\dfrac1{\log_ba} \) and \( \log_{a^m}N=\dfrac1m\log_aN \).
Also \( a^{\log_bc}=c^{\log_ba} \). Converting every logarithm to one base is usually the first step in solving a logarithmic equation.

Sign and Monotonicity

If the base is greater than 1, \( \log_ax \) is increasing: \( \log_ax>\log_ay\Leftrightarrow x>y>0 \). If the base lies between 0 and 1, it is decreasing and inequalities reverse.
For \( \log_xf(x)>1 \) with variable base, treat \( x>1 \) and \( 0<x<1 \) as separate cases, remembering the domain \( x>0,\ x\ne1,\ f(x)>0 \).

Equations and Common Mistakes

After combining logarithms into a single equation, drop the logs only when the bases are equal and then check every solution against the domain of each original logarithm; combining terms can enlarge the domain.
Expressions like \( t+\dfrac1t \) with \( t=\log_ab \) have values with \( |{\cdot}|\ge2 \) by A.M.-G.M., which settles many 'possible values' questions without solving.