Write as a single log : 3log2+2log8- 1/2 log4 ( simplify your answer)
Ans
3log2+2log8- 1/2 log4
= log (2^3) + log(8^2) - log(4^(1/2))
= log 8 + log 64 - log 2
= log(8*64/2)
= log 256
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On the second line of this equation how did you get = log 8 + log 64 - log 2
To simplify the expression 3log2 + 2log8 - 1/2log4, we can use the following properties of logarithms:
1. Logarithm Property: log base b (a) + log base b (c) = log base b (a * c)
2. Logarithm Property: log base b (a) - log base b (c) = log base b (a / c)
3. Logarithm Property: log base b (a^n) = n * log base b (a)
Using these properties, let's simplify the given expression step by step:
Step 1: Apply the logarithm properties to expand the terms:
3log2 + 2log8 - 1/2log4
= log2^3 + log8^2 - log4^(1/2)
Step 2: Simplify the exponents:
= log8 + log64 - log2
= log(8 * 64) - log2
= log(512) - log2
Step 3: Apply the logarithm property to combine the terms:
= log(512/2)
= log(256)
Therefore, the simplified expression is log(256).