Log4 (4 7/17) - log4 (3 21/68) + log4 (48)
Log4 (4 7/17) - log4 (3 21/68) + log4 (48)=3
This is not a true statement. From the solution above, we have:
log4 (4 7/17) - log4 (3 21/68) + log4 (48) ≈ 1.20
This means that the value of the expression is approximately equal to 1.20, not 3. There is no way to manipulate the logarithms to make the expression equal to 3.
To calculate the expression log4 (4 7/17) - log4 (3 21/68) + log4 (48), we can use the properties of logarithms.
Step 1: Simplify each logarithmic expression individually.
The first expression, log4 (4 7/17), can be simplified by converting the mixed number to an improper fraction:
4 7/17 = (4 * 17 + 7) / 17 = 75/17
So, log4 (4 7/17) becomes log4 (75/17).
The second expression, log4 (3 21/68), can also be simplified:
3 21/68 = (3 * 68 + 21) / 68 = 225/68
So, log4 (3 21/68) becomes log4 (225/68).
Lastly, log4 (48) does not require any further simplification.
Step 2: Apply the logarithmic properties.
Using the property loga (b) - loga (c) = loga (b/c), we can simplify the expression.
log4 (75/17) - log4 (225/68) + log4 (48)
= log4 ((75/17) / (225/68)) + log4 (48)
= log4 ((75/17) * (68/225)) + log4 (48)
= log4 (5100/3825) + log4 (48)
= log4 (34/25) + log4 (48)
Step 3: Evaluate the logarithmic expressions.
Using the property loga (b^c) = c * loga (b), we can evaluate each logarithmic expression.
log4 (34/25) can be rewritten as log4 ((2^5 * 34) / (2^5 * 25)):
= log4 (2^5 * 34) - log4 (2^5 * 25)
= 5 * log4 (2) + log4 (34) - 5 * log4 (2) - log4 (25)
= log4 (34) - log4 (25)
Similarly, log4 (48) can be rewritten as log4 (2^4 * 3):
= 4 * log4 (2) + log4 (3)
Now we have:
(log4 (34) - log4 (25)) + (4 * log4 (2) + log4 (3))
Step 4: Combine like terms.
(log4 (34) - log4 (25)) + (4 * log4 (2) + log4 (3))
= log4 (34/25) + 4 * log4 (2) + log4 (3)
And that is the final expression.
Using the logarithmic identity log a (b/c) = log a (b) - log a (c):
log4 (4 7/17) - log4 (3 21/68) + log4 (48)
= log4 [(4×17+7)/17] - log4 [(3×68+21)/68] + log4 (48)
= log4 (75/17) - log4 (213/68) + log4 (48)
Now, we can simplify the logarithms further by using the fact that log a (bc) = log a (b) + log a (c) and log a (b^c) = c log a (b):
= log4 (75) - log4 (17) - [log4 (213) - log4 (68)] + log4 (16×3)
= log4 (75) - log4 (17) - log4 (213) + log4 (68) + log4 (16) + log4 (3)
= log4 [(75×68×16)/(17×213×3)]
≈ log4 (4.23)
≈ 1.20 (rounded to two decimal places)
Therefore, log4 (4 7/17) - log4 (3 21/68) + log4 (48) ≈ 1.20.