For a>0, a 1 and b,c>0, log_a b-log_a c=log_a(bc) Example ln2-ln5=ln 25 log_3 2-log_3 20=log_3 110 log_2 16-log_2 5+log_2 2=log_2(16 x 25) Take note The logarithms being combined must have the same base. The following common mistakes are not identities, even though equality may hold for particular allowed values. The claim log_a(p-q)= log_a plog_a q is not an identity. The claim log_a(p+q)=log_a p+log_a q is not an identity. Check that every logarithm argument is positive and every denominator is nonzero.

\(\log_a b-\log_a c=\log_a\left(\dfrac{b}{c}\right)\)

\(\log_2 16-\log_2 5+\log_2 2=\log_2\left(\dfrac{16\times2}{5}\right)\)

The logarithms being combined must have the same base. The following common mistakes are not identities, even though equality may hold for particular allowed values.

The claim \(\log_a(p-q)=\dfrac{\log_a p}{\log_a q}\) is not an identity.

The claim \(\log_a(p+q)=\log_a p+\log_a q\) is not an identity.

Check that every logarithm argument is positive and every denominator is nonzero.

Exponential and Logarithmic Functions

International Baccalaureate Mathematics: Analysis and Approaches Higher Level

Quotient Law

Quotient Law