JEE Mathematics Logarithms Notes Set 01

Definition and Basic Facts

1. \(\log_aN=x\) means \(a^x=N\). It exists only when \(N>0\), \(a>0\) and \(a\ne1\).
2. \(\log_a1=0\), \(\log_aa=1\), \(\log_{1/N}N=-1\), and \(a^{\log_aN}=N\).
3. If the number and the base lie on the same side of 1 the logarithm is positive; if they lie on opposite sides it is negative.
4. Antilog: \(\operatorname{antilog}_b x=b^x\).

Laws of Logarithms

Be careful: the laws are valid only when all arguments are positive. For example \(\log_2x^2=4\) has two solutions \(x=\pm4\), while \(2\log_2x=4\) has only \(x=4\); use \(\log_ax^2=2\log_a|x|\).

Solving Logarithmic and Exponential Equations

1. First write the domain: every argument positive, every base positive and not 1.
2. Reduce to a single logarithm on each side, or use a substitution such as \(t=\log_ax\) to get a quadratic.
3. For equations like \(x^{f(x)}=c\), take the logarithm of both sides to get a polynomial in \(\log x\).
4. For \(a^{f}=1\), check the three cases: base 1, exponent 0 with non-zero base, and base -1 with even exponent.
5. Always test the roots against the domain; extraneous roots are common.

Inequalities

1. If \(a>1\): \(\log_ax\lt\log_ay\iff0\lt x\lt y\).
2. If \(0\lt a\lt 1\): \(\log_ax\lt\log_ay\iff x>y>0\), so the inequality reverses.

Characteristic and Mantissa

1. Write \(\log_bN=\alpha+\beta\) with integer \(\alpha\) (characteristic) and \(0\le\beta<1\) (mantissa).
2. Then \(b^\alpha\le N3. For base 10, the number of digits of N (with \(N\ge1\)) is \(\alpha+1\).
4. If \(\log_{10}N=-k+\beta\) with \(k\ge1\), then N has \(k-1\) zeros after the decimal point before the first significant digit.