Question

If log 10 a=x, log 10 b=y and log 10 c=z. Express P in terms of a, b and c if log 10 P=3x-5y+2z

Answer 1

$given \: log_{a}10 = x \: and \: log_{b}10 = y \: \: and \: log_{c}10 = z \\ \\ = > \frac{1}{x} = log_{10}(a) \: \: \: and \: \: log_{10}(b) = \frac{1}{y} \: \: and \: log_{c}(10 ) = \frac{1}{c} \\ \\ = > {3x - 5y + 2z} \\ \\ = \frac{1}{3 log_{10}(a) - 5log_{10}(b) + 2 log_{10}(c) } \\ \\ = \frac{1}{ log_{10}(a {}^{3} ) - log_{10}(b {}^{5} ) + log_{10}(c {}^{2} ) } \: \: \: ( log_{b}(a {}^{m} ) = m log_{b}(a) ) \\ \\ = \frac{1}{ log_{10}(a {}^{3}c {}^{2} ) - log_{10}(b {}^{5} ) } \: \: \: \: ( log_{b}(a) + log_{b}(c) = log_{b}(ac) ) \\ \\ = \frac{1}{ log_{10}( \frac{a {}^{3}c {}^{2} }{b {}^{5} } ) } \: \: ( log_{c}(a) - log_{c}(b) = log_{c}( \frac{a}{b} ) ) \\ \\ = log_{ \frac{a {}^{3}c {}^{2} }{b {}^{5} } }(10) \\ \\ given \: log_{10}(p) = 3x - 5y + 2z \\ \\ = > log_{10}(p) = \frac{1}{ log_{10}( \frac{a {}^{3}c {}^{2} }{b {}^{5} } ) } \\ \\ since \: bases \: are \: equal \: \\ \\ = > p = \frac{1}{ \frac{a {}^{3}c {}^{2} }{b {}^{5} } } \\ \\ = > p = \frac{b {}^{5} }{a {}^{3}c {}^{2} }$