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Question Number 221896 by fantastic last updated on 12/Jun/25

If log _(10) 7=a ,then log _(10) ((1/(70)))=?

$${If}\:\mathrm{log}\underset{\mathrm{10}} {\:}\mathrm{7}={a}\:,{then}\:\mathrm{log}\underset{\mathrm{10}} {\:}\left(\frac{\mathrm{1}}{\mathrm{70}}\right)=? \\ $$

Answered by fantastic last updated on 12/Jun/25

log _(10) 7=a so 10^a =7 or 70 =10^a .10 =10^((a+1)) So (1/(70)) =(1/(10^((a+1)) )) =10^(−(a+1)) then log _(10) ((1/(70))) =log _(10) 10^(−(a+1)) =−(a+1)log _(10) 10 =−(a+1)×1 =−(a+1) am I wrong???

$$\mathrm{log}\underset{\mathrm{10}} {\:}\mathrm{7}={a} \\ $$$$\:{so}\:\mathrm{10}^{{a}} =\mathrm{7}\: \\ $$$${or}\:\mathrm{70}\:=\mathrm{10}^{{a}} .\mathrm{10} \\ $$$$=\mathrm{10}^{\left({a}+\mathrm{1}\right)} \\ $$$${So}\:\frac{\mathrm{1}}{\mathrm{70}} \\ $$$$=\frac{\mathrm{1}}{\mathrm{10}^{\left({a}+\mathrm{1}\right)} } \\ $$$$=\mathrm{10}^{−\left({a}+\mathrm{1}\right)} \\ $$$${then} \\ $$$$\:\mathrm{log}\underset{\mathrm{10}} {\:}\left(\frac{\mathrm{1}}{\mathrm{70}}\right) \\ $$$$=\mathrm{log}\underset{\mathrm{10}} {\:}\mathrm{10}^{−\left({a}+\mathrm{1}\right)} \\ $$$$=−\left({a}+\mathrm{1}\right)\mathrm{log}\underset{\mathrm{10}} {\:}\mathrm{10} \\ $$$$=−\left({a}+\mathrm{1}\right)×\mathrm{1} \\ $$$$=−\left({a}+\mathrm{1}\right) \\ $$$${am}\:{I}\:{wrong}??? \\ $$

Commented by mahdipoor last updated on 12/Jun/25

nope , log(1/(70))=−log70=−(log10+log7)=−(1+a)

$${nope}\:, \\ $$$${log}\frac{\mathrm{1}}{\mathrm{70}}=−{log}\mathrm{70}=−\left({log}\mathrm{10}+{log}\mathrm{7}\right)=−\left(\mathrm{1}+{a}\right) \\ $$

Commented by fantastic last updated on 12/Jun/25

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