Math Snap

STEP 1

What is this asking?
We need to rewrite the natural logarithm of a fraction with variables and a constant into a simpler form using logarithm rules.
Watch out!
It's easy to mess up the signs when using the logarithm rules, so let's be extra careful!

STEP 2

1. Expand the numerator
2. Expand the denominator
3. Combine the results

STEP 3

We have $\ln\frac{2a^3}{b^4}$.
We can rewrite the numerator using the product rule for logarithms, which says $\ln(xy) = \ln(x) + \ln(y)$.
So, $\ln(2a^3)$ becomes $\ln(2) + \ln(a^3)$.
This rule helps us break down complex expressions inside the logarithm!

STEP 4

Now, let's use the power rule for logarithms, which says $\ln(x^y) = y\cdot \ln(x)$.
Applying this to $\ln(a^3)$, we get $3\cdot \ln(a)$.
So, the numerator becomes $\ln(2) + 3\cdot\ln(a)$.
This rule helps us bring exponents down as coefficients.

STEP 5

The denominator is $\ln(b^4)$.
Using the power rule for logarithms, which says $\ln(x^y) = y\cdot \ln(x)$, we get $4\cdot \ln(b)$.

STEP 6

Our expression is now $\ln\frac{2a^3}{b^4} = \ln(2a^3) - \ln(b^4)$, which we've already simplified to $\ln(2) + 3\cdot\ln(a) - 4\cdot\ln(b)$.

STEP 7

We've used the logarithm rules to expand everything correctly!
Our final expression is $\ln(2) + 3\cdot\ln(a) - 4\cdot\ln(b)$.

The expanded form of $\ln \frac{2 a^{3}}{b^{4}}$ is $\ln 2+3 \ln a-4 \ln b$.