$\"\"$

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Prove the quotient property of logarithm using graphs.

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Quotient property of logarithm is : $\"\"$ .

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Consider the logarithm expression $\"\"$ .

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Apply quotient property of logarithm.

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$\"\"$ .

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Graph :

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Graph of the logarithm expression $\"\"$ .

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$\"\"$

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Graph of the logarithm expression $\"\"$ .

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$\"\"$

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Observe the two graphs both are same, so the Quotient property of logarithm proved.

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$\"\"$

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Prove the quotient property of logarithm using equations.

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Consider two equations $\"\"$ and $\"\"$ .

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$\"\"$

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Apply the definition of logarithm is : $\"\"$ if and only if $\"\"$ .

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$\"\"$ .

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$\"\"$

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Apply the definition of logarithm is : $\"\"$ if and only if $\"\"$ .

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$\"\"$ .

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$\"\"$

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Rewrite the above expression.

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$\"\"$

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Apply power property of logarithm : $\"\"$ .

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$\"\"$

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Apply property of logarithm : $\"\"$ .

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$\"\"$

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Substitute $\"\"$ and $\"\"$ in the above expression.

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$\"\"$ .

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$\"\"$ .

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$\"\"$

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The quotient property of logarithm $\"\"$ .