Express ^@(log _a \space x)(log _b \space a)^@ as a single logarithm.
Answer:
^@log _b \space x^@
- According to the change of base formula of logarithm, ^@log _b \space m = \dfrac{ log _a \space m } { log _a \space b }^@
- We can write ^@(log _a \space x)^@ as ^@\dfrac{ log _b \space x }{ log _b \space a } ^@
- ^@(log _a \space x)(log _b \space a) ^@^@=
\left( \dfrac{ log _b \space x } { log _b \space a } \right)
(log _b \space a)^@
^@\implies (log _a \space x)(log _b \space a) = log _b \space x.^@
Hence, ^@(log _a \space x)(log _b \space a) ^@ as a single logarithm is ^@log _b \space x^@.