simplify 2log4(9)-log2(3)
To simplify the expression 2log4(9) - log2(3), we can use the properties of logarithms.
First, let's simplify the each individual logarithm using the change of base formula:
log4(9) = log(9) / log(4)
log2(3) = log(3) / log(2)
Now, let's substitute these simplified logarithms back into the original expression:
2log4(9) - log2(3) = 2 * (log(9) / log(4)) - (log(3) / log(2))
Next, we'll use the property of logarithms that states log(a) - log(b) = log(a/b):
2 * (log(9) / log(4)) - (log(3) / log(2)) = log(9^2) / log(4) - log(3) / log(2)
Simplifying further:
= log(81) / log(4) - log(3) / log(2)
Since log(81) can be expressed as log(3^4), we can rewrite it as:
= log(3^4) / log(4) - log(3) / log(2)
Using the power property of logarithms log(a^b) = b * log(a), we get:
= 4 * log(3) / log(4) - log(3) / log(2)
Now, we need a common denominator for log(4) and log(2), which is log(2^2). We can rewrite it as:
= 4 * log(3) / log(4) - log(3) / (2 * log(2))
Combining the terms:
= (4 * log(3) - log(3)) / (log(4) - 2 * log(2))
Simplifying the numerator:
= (3 * log(3)) / (log(4) - 2 * log(2))
Finally, let's simplify the denominator:
= (3 * log(3)) / (log(4) - log(2^2))
Using the quotient rule of logarithms log(a) - log(b) = log(a/b), we get:
= (3 * log(3)) / log(4/2^2)
Simplifying the denominator further:
= (3 * log(3)) / log(4/4)
Since 4/4 equals 1, we end up with:
= 3 * log(3) / log(1)
The logarithm of 1 is always zero: log(1) = 0. Therefore, the simplified form is:
= 3 * log(3) / 0
However, dividing by zero is undefined, so the expression is undefined.
To simplify the expression 2log4(9) - log2(3), we can use logarithmic properties.
1. First, use the power rule for logarithms: loga(b^c) = c*loga(b). Applying this rule to the expression, we have:
2log4(9) = log4(9^2) = log4(81)
2. Next, simplify log4(81) using the change of base formula:
log4(81) = log(81) / log(4)
Note that log can be any base, commonly used bases are log(10) or ln (log base e).
3. Similarly, we need to simplify log2(3) using the change of base formula:
log2(3) = log(3) / log(2)
4. Now we can rewrite the expression using the simplified forms:
2log4(9) - log2(3) = log(81) / log(4) - log(3) / log(2)
We can leave it at this step or further simplify by finding a common denominator:
[log(81) * log(2) - log(3) * log(4)] / [log(4) * log(2)]
This is the simplified form of the expression 2log4(9) - log2(3).
2log4(9)-log2(3) = 2log2(9^(1/2))-log2(3)
= 2log2(3)-log2(3)
= log2(3)