Globokim8

2021-08-11

Use properties of logarithms to condense the logarithmic expression log 5 + log 2. Write the expression as a single logarithm whose coefficient is 1. Where possible, evaluate logarithmic expressions without using a calculator.

BleabyinfibiaG

We have to condense the logarithmic expression as well as we have to find the exact value where expression is:
$\mathrm{log}\left(5\right)+\mathrm{log}\left(2\right)$
We know that for general logarithm there is base 10.
So rewriting the given logarithmic expression,
$\mathrm{log}\left(5\right)+\mathrm{log}\left(2\right)={\mathrm{log}}_{10}\left\{5\right\}+{\mathrm{log}}_{10}\left\{2\right\}$
We know properties of logarithm,
$\mathrm{log}\left(a\right)+\mathrm{log}\left(b\right)\mathrm{log}\left(ab\right)$
$=\mathrm{log}\left(ab\right)1$
Applying above property for the given expression, we get
$\mathrm{log}\left(a\right)+\mathrm{log}\left(b\right)$
$=\mathrm{log}\left(5\right)+\mathrm{log}\left(2\right)\mathrm{log}\left(ab\right)\mathrm{log}\left(5×2\right)$
$=\mathrm{log}\left(10\right)$
Hence, condense expression of logarithm is $\mathrm{log}\left(10\right)$.
If base of logarithm is 10 then expression value will be
${\mathrm{log}}_{10}\left\{5\right\}+{\mathrm{log}}_{10}\left\{2\right\}={\mathrm{log}}_{10}\left\{5×2\right\}$
${\mathrm{log}}_{10}\left\{10\right\}$
=1

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