Use the properties of logarithms to rewrite and simplify the logarithmic expression.
1.) log(9/300)
2.) log2(4^2 x 3^4)
1. Log(9/300) = Log 9 - Log 300
2. Log2(4^2*3^4) = Log2(4^2) + Log2(3^4)
= Log2(16) + Log2(81)
1.) To rewrite and simplify the logarithmic expression log(9/300), we can use the property of logarithms that states log(a/b) is equal to log(a) - log(b).
Therefore, log(9/300) = log(9) - log(300).
Now, we can simplify further by using the property of logarithms that states log(a^b) is equal to b * log(a).
So, log(9) - log(300) = log(3^2) - log(300) = 2 * log(3) - log(300).
And finally, we can simplify the expression by using the property of logarithms that states log(a * b) is equal to log(a) + log(b).
So, 2 * log(3) - log(300) = log(3^2) + log(3^-2) - log(300) = log(3^2 * 3^-2) - log(300).
The simplified logarithmic expression is log(1/100) - log(300).
To rewrite and simplify logarithmic expressions using the properties of logarithms, we can make use of the following properties:
1. The logarithm of a quotient is equal to the logarithm of the numerator minus the logarithm of the denominator.
```log(a/b) = log(a) - log(b)```
2. The logarithm of a power is equal to the exponent multiplied by the logarithm of the base.
```log(a^b) = b * log(a)```
Now, let's apply these properties to the given expressions:
1. log(9/300)
Apply the quotient property, log(a/b) = log(a) - log(b):
```= log(9) - log(300)```
Now, you can simplify the expression further by finding the logarithm values of 9 and 300 using the base of your logarithm function (e.g., base 10 or base e).
2. log2(4^2 x 3^4)
Apply the power property, log(a^b) = b * log(a):
```= 2 * log2(4) + 4 * log2(3)```
Since log2(4) and log2(3) are likely not values you know off the top of your head, you can use a calculator or mathematical software to evaluate each logarithm expression, and then perform the multiplications.
Remember, when simplifying logarithmic expressions, it's important to use the properties of logarithms and evaluate the logarithm values if necessary to achieve the final simplified form.