jhenezhubby01ff

2022-09-07

Need to simplify a logarithmic expression

Can someone simplify this ($\mathrm{log}$ here refers to the common logarithm)?

$\sqrt{4\mathrm{log}2+(\mathrm{log}5{)}^{2}}+\sqrt{4\mathrm{log}5+(\mathrm{log}2{)}^{2}}$

I know this has a simple solution but I cannot find it.

Can someone simplify this ($\mathrm{log}$ here refers to the common logarithm)?

$\sqrt{4\mathrm{log}2+(\mathrm{log}5{)}^{2}}+\sqrt{4\mathrm{log}5+(\mathrm{log}2{)}^{2}}$

I know this has a simple solution but I cannot find it.

Samantha Braun

Beginner2022-09-08Added 9 answers

It seems pretty sneaky. So I'm guessing those are common logs. Let's take a look at the first square root. We have

$\sqrt{4\mathrm{log}2+(\mathrm{log}5{)}^{2}}=\sqrt{4(\mathrm{log}10-\mathrm{log}5)+(\mathrm{log}5{)}^{2}}=\sqrt{4-4\mathrm{log}5+(\mathrm{log}5{)}^{2}}=2-\mathrm{log}5$

Similarly, the second square root yields $2-\mathrm{log}2$. Sum them together to get $4-\mathrm{log}10=3$

$\sqrt{4\mathrm{log}2+(\mathrm{log}5{)}^{2}}=\sqrt{4(\mathrm{log}10-\mathrm{log}5)+(\mathrm{log}5{)}^{2}}=\sqrt{4-4\mathrm{log}5+(\mathrm{log}5{)}^{2}}=2-\mathrm{log}5$

Similarly, the second square root yields $2-\mathrm{log}2$. Sum them together to get $4-\mathrm{log}10=3$

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