Use the properties of logarithms to find the exact value of the expression log[16]4*log[5]sqrt7?
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To find the exact value of the expression log[16]4 * log[5]√7, we can use the properties of logarithms.
First, let's rewrite the expression using the properties of logarithms:
log[16]4 = log[16](2^2) = 2*log[16]2
log[5]√7 = log[5](7^(1/2)) = (1/2)*log[5]7
Now, we can substitute these expressions back into the original expression:
log[16]4 * log[5]√7 = (2*log[16]2) * ((1/2)*log[5]7)
Using the property log[a]b^c = c * log[a]b, we can simplify further:
= 2 * log[16]2 * (1/2) * log[5]7
= log[16]2 * log[5]7
Next, we can use the property log[a]b = log[c]b / log[c]a to simplify even more:
= log[2]2 / log[2]16 * log[7]7 / log[7]5
Finally, we can simplify these logarithms further:
- log[2]16 = log[2](2^4) = 4
- log[7]7 = 1
- log[7]5 cannot be further simplified.
Now, we substitute these values back into the expression:
= log[2]2 / 4 * 1 / log[7]5
= 1 / 4 * 1 / log[7]5
Therefore, the exact value of the expression log[16]4 * log[5]√7 is 1 / 4 * 1 / log[7]5.