Write each expression as a sum or difference of logarithms. Express powers as factors.
log2(x^3/x - 3)
x > 3
Answer: 3 log2x - log2 (x-3)
Could someone show me the steps to get to this answer?
Review the basic rules of logarithms.
log(a/b) = loga - logb
log(x^n) = n * log(x)
Sure! To express the given expression as a sum or difference of logarithms, we first need to simplify it. Let's break it down into two separate parts:
1) log2(x^3) - log2(x - 3)
Now, let's simplify each part separately:
1) Simplifying log2(x^3):
Using the logarithmic property, loga(b^c) = c * loga(b), we can rewrite log2(x^3) as 3 * log2(x):
3 * log2(x)
2) Simplifying log2(x - 3):
No further simplification can be done on this part, so we leave it as it is:
log2(x - 3)
Now, we can combine the simplified parts back together:
3 * log2(x) - log2(x - 3)
And there you have it! The given expression has been expressed as a sum or difference of logarithms: 3 * log2(x) - log2(x - 3).
To express the expression log2(x^3/x - 3) as a sum or difference of logarithms, we can apply the quotient rule of logarithms.
The quotient rule states that log base b of (a/c) is equal to log base b of a minus log base b of c.
So, let's break down the given expression:
log2(x^3/x - 3)
First, we can rewrite the expression as the difference of two logarithms using the quotient rule:
= log2(x^3) - log2(x - 3)
Next, we can use the power rule of logarithms, which states that log base b of a to the power of n is equal to n times log base b of a.
In this case, we have log2(x^3), so we can rewrite it as:
= 3log2(x) - log2(x - 3)
Therefore, the expression log2(x^3/x - 3) can be expressed as 3log2(x) - log2(x - 3).