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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

Skilled2021-08-11Added 94 answers

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$\mathrm{log}}_{4}\frac{13}{a$

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

The power rule of logarithm is: a>0, a ne 1, and any exponent value n,

The quotient rule of the logarithm is for any base a>0, a ne 1

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

The answer is

$\frac{20b}{{\left(4{b}^{3}\right)}^{3}}$

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