The examples below will teach you about expanding logarithms using the properties of logarithms.
Expanding logarithms is the inverse of simplifying logarithms. You are trying to express a single logarithm into many individual logarithms.
Example #1
Expand log_{5} (3xy)
Using the product property, log_{5} (3xy) = log_{5} 3 + log_{5} x + log_{5} y
Example #2
Expand log_{2} (5x / 10y)
Using a combination of the product property and the quotient property,
log_{2} (5x / 10y) = log_{2} (5x) - log_{2} (10y)
= log_{2} (5) + log_{2} (x) - [ log_{2} (10) + log_{2} (y) ]
= log_{2} (5) + log_{2} (x) - log_{2} (10) - log_{2} (y)
Example #3
Expand log (a^{2}b^{3})
Using a combination of the product property and the power property,
log_{10} (a^{2}b^{3}) = log_{10} a^{2} + log_{10} b^{3}
= 2log_{10} a + 2log_{10}
Example #4
Expand log3 [ √ (36x^{3}y) ]
Using a combination of the product property and the power property,
log3 [ √ (36x^{3}y) ] = log3 [ (36x^{3}y)^{1/2} ]
= 1/2 log3 (36x^{3}y)
= 1/2 [ log3 (36) + log3 (x^{3}) + log3 (y) ]
= 1/2 [ log3 (36) + 3log3 (x) + log3 (y) ]
= 1/2 log3 (36) + 3/2 log3 (x) + 1/2 log3 (y)
Dec 12, 19 07:51 AM
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