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How can we write logarithm of 18 to the base 3 in terms of logarithm of 12 to the base 3?

Answer 1

Benjamin Achtenberg

#log_3 18 = log_3 12 + log_3 (3/2)#

Since #log_b (p * q) = log_b p + log_b q#

#log_3 18 = log_3 (12 * 3/2)#

#color(white)("XXX")=log_3 (12) + log_3 (3/2)#

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Answer 2

Lincoln Anderson

We can write the logarithm of 18 to the base 3 in terms of the logarithm of 12 to the base 3 using the property of logarithms:

[ \log_3(18) = \log_3(2 \times 9) = \log_3(2) + \log_3(9) ]

[ = \log_3(2) + \log_3(3^2) = \log_3(2) + 2\log_3(3) ]

Now, since we want to express this in terms of the logarithm of 12 to the base 3, we note that (12 = 2 \times 6):

[ \log_3(12) = \log_3(2 \times 6) = \log_3(2) + \log_3(6) ]

[ = \log_3(2) + \log_3(2 \times 3) = \log_3(2) + \log_3(2) + \log_3(3) = 2\log_3(2) + \log_3(3) ]

So, substituting this expression into our original one:

[ \log_3(18) = 2\log_3(12) - \log_3(3) ]

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.