Applying the Log-Exponent Relationship
Keeping the log-exponent relationship in mind can help us rewrite logarithmic expressions and evaluate them. Exponents and logarithms are inverse operations, and we can rewrite these two kinds of expressions in the following way:
Consider the following expression:
We can apply various properties of logs to simplify and evaluate this expression. First, let's use the inverse relationship between exponents and logs to think about how to evaluate each logarithmic term individually:
We can now rewrite our original expression as: which we know is
Applying the Product and Quotient Properties
Let's simplify and evaluate the same expression, but this time we will use the product and quotient properties of logarithms.
In our first example, we have some logarithms with the same base being added and subtracted. When we see addition, we can combine them into one logarithm by multiplying the arguments. Similarly, when we see subtraction, we can combine them into one logarithm using division.
Applying Multiple Properties
What kinds of properties would we use to simplify and evaluate the following expression?
Inside the logarithm, we see multiplication, division, and an exponent. This tells us that we will likely use the Product, Quotient, and Power Properties. We are also given the values of log(x) and log(y). However, notice the difference in bases. The expression we need to evaluate has a log of base 2, whereas the log values we are given are common logs, which have a base of 10. This means we'll need to use the Change of Base Property as well.