Math  /  Algebra

Questiond) log(xy)+logyx\log (x y)+\log \frac{y}{x}

Studdy Solution

STEP 1

What is this asking? Simplify this logarithmic expression by combining the two logarithmic terms. Watch out! Remember the logarithmic properties, especially the product and quotient rules!
Don't forget to consider any potential restrictions on the variables.

STEP 2

1. Expand the Expression
2. Combine Like Terms

STEP 3

Alright, let's **expand** that first term, log(xy)\log(xy), using the **product rule** of logarithms.
Remember, the product rule says log(ab)=log(a)+log(b)\log(a \cdot b) = \log(a) + \log(b).
So, log(xy)\log(xy) becomes log(x)+log(y)\log(x) + \log(y).
This is because xx multiplied by yy inside the logarithm can be split into the sum of two separate logarithms.

STEP 4

Now, let's tackle the second term, logyx\log \frac{y}{x}.
We'll use the **quotient rule** here, which states log(ab)=log(a)log(b)\log(\frac{a}{b}) = \log(a) - \log(b).
Applying this to our term gives us log(y)log(x)\log(y) - \log(x).
This is because yy divided by xx inside the logarithm can be split into the difference of two separate logarithms.

STEP 5

So, our original expression log(xy)+logyx\log(xy) + \log\frac{y}{x} now looks like this: log(x)+log(y)+log(y)log(x)\log(x) + \log(y) + \log(y) - \log(x).
Much better, right?

STEP 6

Let's **group** our like terms together.
We've got log(x)\log(x) and log(x)-\log(x), and also log(y)\log(y) and log(y)\log(y).
So, we can rewrite our expression as (log(x)log(x))+(log(y)+log(y))(\log(x) - \log(x)) + (\log(y) + \log(y)).

STEP 7

Notice that log(x)log(x)\log(x) - \log(x) adds to **zero**.
Awesome! That simplifies things.
We are left with log(y)+log(y)\log(y) + \log(y).

STEP 8

Since we have two log(y)\log(y) terms, we can rewrite this as 2log(y)2\log(y).
Think of it like having one log(y)\log(y) plus another log(y)\log(y), which gives us a total of two log(y)\log(y)!

STEP 9

We can also use the **power rule** of logarithms, which says nlog(a)=log(an)n\log(a) = \log(a^n), to rewrite 2log(y)2\log(y) as log(y2)\log(y^2).
This step is optional, but it's good to know!

STEP 10

Our simplified expression is 2log(y)2\log(y), or equivalently, log(y2)\log(y^2).

Was this helpful?

Studdy solves anything!

banner

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

ParentsInfluencer programContactPolicyTerms
TwitterInstagramFacebookTikTokDiscord