If $x = log 2$ $x = log2$ and $y = log 3$ $y = log3$ , express $log 75$ $log75$ in terms of $x$ $x$ and $y$ $y$ ?

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All bases are 10.

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Answer 1 · 2017-09-14T13:31:04

${log}_{10} 75 = 2 - 2 x + y .$ $log_(10)75=2-2x+y.$

Given that, ${log}_{10} 2 = x, and, {log}_{10} 3 = y .$ $log_(10)2=x, and, log_(10)3=y.$

We will use, in what follows, the familiar Rules of Log. Fun.

We have,

${log}_{10} 75 = {log}_{10} (3 \times 5^{2}),$ $log_(10)75=log_(10)(3xx5^2),$

$= {log}_{10} 3 + {log}_{10} 5^{2},$ $=log_(10)3+log_(10)5^2,$

$=y+2log_(10)5..........................[because, log_(10)3=y," given],"$

$=y+2log_(10)(10-:2),$

$=y+2{log_(10)10-log_(10)2},$

$=y+2(1-x)............................[because, log_(10)2=x," given]."$

$rArr log_(10)75=2-2x+y.$