Write the expression as a sum and/or difference of logarithms. Express powers as factors.
log3(27x)
log3(27)+log3(x)
3+log3(x)
Log(3)xy
To express the logarithmic expression log3(27x) as a sum and/or difference of logarithms, we will need to use the properties of logarithms. Specifically, we will use the property that allows us to write a logarithm of a product as a sum of logarithms.
The property states that log(base a)(b * c) = log(base a)(b) + log(base a)(c)
Applying this property to the given expression log3(27x), we can rewrite it as:
log3(27) + log3(x)
Now, let's simplify further. The value of log3(27) is the exponent we need to raise 3 to in order to get 27. Since 3^3 = 27, we have:
log3(27) = 3
Now, the expression becomes:
3 + log3(x)
Therefore, the expression log3(27x) can be written as a sum of logarithms:
3 + log3(x)
To express the logarithmic expression log3(27x) as a sum and/or difference of logarithms, we need to break down the argument (the value inside the logarithm) into factors or powers. In this case, the argument is 27x.
Using the properties of logarithms, we can rewrite 27 as a power of 3 since log3(27) is meaningful and equal to 3. Similarly, we can rewrite x as a separate factor since log3(x) is meaningful.
Therefore, we can rewrite the given logarithmic expression as a sum of logarithms as follows:
log3(27x) = log3(27) + log3(x)
Since 27 is equal to 3 raised to the power of 3 (i.e., 3^3), and x is not a power of 3, the expression becomes:
log3(27x) = log3(3^3) + log3(x)
Using the properties of logarithms, we can simplify further:
log3(27x) = 3log3(3) + log3(x)
Finally, we simplify the logarithm of the base:
log3(27x) = 3(1) + log3(x)
Therefore, the expression log3(27x) can be written as the sum of logarithms as:
log3(27x) = 3 + log3(x)