Write as a single logarithm:
2 log 3 – log 5 +2 log2
log 9 - log 5 + log 4 = log 36/5
n * log ( a ) = log ( a ^ n )
log ( a ) + log ( b ) = log ( a * b )
log ( a ) - log ( b ) = log ( a / b )
2 log ( 3 ) = log ( 3 ^ 2 ) = log ( 9 )
2 log ( 2 ) = log ( 2 ^ 2 ) = log ( 4 )
2 log ( 3 ) - log ( 5 ) + 2 log ( 2 ) =
log ( 3 ^ 2 * 2 ^ 2 / 5 ) =
log ( 9 * 4 / 5 ) =
log ( 36 / 5 )
To write the expression as a single logarithm, we can combine the logarithmic terms using the properties of logarithms.
Using the property log(a) + log(b) = log(ab), we can rewrite the expression as:
2 log 3 – log 5 + 2 log 2 = log(3^2) - log(5) + log(2^2)
Simplifying further, we get:
= log(9) - log(5) + log(4)
Now, applying the property log(a) - log(b) = log(a/b), we have:
= log(9/5) + log(4)
Finally, using the property log(a) + log(b) = log(ab), we can rewrite the expression as a single logarithm:
= log(9/5 * 4)
Therefore, the expression can be written as a single logarithm: log(36/5).
To write the given expression as a single logarithm, we can use the properties of logarithms. The properties we will use are:
1. Product Rule: log a + log b = log (a * b)
2. Quotient Rule: log a - log b = log (a / b)
3. Power Rule: log a^b = b * log a
Let's simplify the given expression step by step:
2 log 3 – log 5 + 2 log 2
Using the Product Rule, we can rewrite the expression as:
log (3^2) - log 5 + log (2^2)
This simplifies to:
log (9) - log 5 + log (4)
Using the Quotient Rule, we can rewrite the expression as:
log (9 / 5) + log 4
Finally, using the Product Rule, we can combine the logarithms:
log [(9 / 5) * 4]
Simplifying further:
log (36 / 5)
Therefore, the given expression can be written as a single logarithm:
log (36 / 5)