QuestionUse the Laws of Logarithms to combine the expression. $\ln (a+b)+\ln (a-b)-2 \ln (c)$

Studdy Solution

STEP 1

What is this asking? Rewrite the sum and difference of these natural logarithms as a single logarithm. Watch out! Remember the logarithm rules and be careful with the signs!

STEP 2

1. Apply the logarithm product rule.
2. Apply the logarithm power rule.
3. Apply the logarithm quotient rule.

STEP 3

Alright, let's combine the first two logarithms!
Since we're adding two logarithms with the same base, we can multiply the arguments inside a single logarithm.
This is the product rule for logarithms!

STEP 4

So, $\ln(a+b) + \ln(a-b)$ becomes $\ln((a+b) \cdot (a-b))$ .
See how we multiplied the $(a+b)$ and $(a-b)$ inside the logarithm?

STEP 5

Let's simplify this! $(a+b) \cdot (a-b)$ is a difference of squares, which simplifies to $a^2 - b^2$ .
So, our expression now looks like $\ln(a^2 - b^2) - 2 \ln(c)$ .
Nice!

STEP 6

Now, let's look at that $2 \ln(c)$ term.
We can use the power rule of logarithms to rewrite this.
The power rule says that $n \cdot \ln(m)$ is the same as $\ln(m^n)$ .

STEP 7

Applying this rule, we can rewrite $2 \ln(c)$ as $\ln(c^2)$ .
So, our expression is now $\ln(a^2 - b^2) - \ln(c^2)$ .
Getting closer!

STEP 8

Finally, we have a difference of two logarithms with the same base.
We can use the quotient rule for logarithms to combine these.
The quotient rule says $\ln(m) - \ln(n)$ is equal to $\ln(\frac{m}{n})$ .

STEP 9

Applying this rule, we can rewrite $\ln(a^2 - b^2) - \ln(c^2)$ as $\ln(\frac{a^2 - b^2}{c^2})$ .
And there we have it!
We've combined the entire expression into a single logarithm.

STEP 10

The combined expression is $\ln(\frac{a^2 - b^2}{c^2})$ .

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