sjeikdom0

2021-08-10

Find the sum, difference, or product of logarithms as a single logarithm, if it is possible, using the properties of logarithms
${\mathrm{log}}_{4}13-{\mathrm{log}}_{4}a$

aprovard

The logarithm is
${\mathrm{log}}_{4}13-{\mathrm{log}}_{4}a$
The objective is to use the property of logarithm to combine the terms in single logarithm
The product rule of the logarithm is: for any base a>0, a ne 1
${\mathrm{log}}_{a}\left(uv\right)={\mathrm{log}}_{a}u+{\mathrm{log}}_{a}v$
The power rule of logarithm is: a>0, a ne 1, and any exponent value n,
${\mathrm{log}}_{a}\left({u}^{n}\right)=n{\mathrm{log}}_{a}u$
The quotient rule of the logarithm is for any base a>0, a ne 1
${\mathrm{log}}_{a}\left(\frac{u}{v}\right)={\mathrm{log}}_{a}u-{\mathrm{log}}_{a}v$
To combine the logarithmic expression into single logarithm, the bases of logarithm must be same
So, the given logarithmic expression is converted into single logarithm using properties of logarithm
${\mathrm{log}}_{4}13-{\mathrm{log}}_{4}a={\mathrm{log}}_{4}\frac{13}{a}$
The answer is ${\mathrm{log}}_{4}\frac{13}{a}$

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