Math / Algebra

QuestionExpand: $\ln \frac{2 a^{3}}{b^{4}}$ $\ln 2+3 \ln a-4 \ln b$ $\ln 2-3 \ln a-4 \ln b$ $\ln 2-3 \ln a+4 \ln b$

Studdy Solution

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 $\ln(2) + 3\cdot\ln(a) - 4\cdot\ln(b)$ .

STEP 8

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

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QuestionExpand: $\ln \frac{2 a^{3}}{b^{4}}$ $\ln 2+3 \ln a-4 \ln b$ $\ln 2-3 \ln a-4 \ln b$ $\ln 2-3 \ln a+4 \ln b$

Studdy Solution

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 $\ln(2) + 3\cdot\ln(a) - 4\cdot\ln(b)$ .

STEP 8

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

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QuestionExpand: ln⁡2a3b4\ln \frac{2 a^{3}}{b^{4}}lnb42a3​ ln⁡2+3ln⁡a−4ln⁡b\ln 2+3 \ln a-4 \ln bln2+3lna−4lnb ln⁡2−3ln⁡a−4ln⁡b\ln 2-3 \ln a-4 \ln bln2−3lna−4lnb ln⁡2−3ln⁡a+4ln⁡b\ln 2-3 \ln a+4 \ln bln2−3lna+4lnb

Studdy Solution

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 numerator2. Expand the denominator3. Combine the results

STEP 3

STEP 4

STEP 5

The denominator is ln⁡(b4)\ln(b^4)ln(b4).Using the power rule for logarithms, which says ln⁡(xy)=y⋅ln⁡(x)\ln(x^y) = y\cdot \ln(x)ln(xy)=y⋅ln(x), we get 4⋅ln⁡(b)4\cdot \ln(b)4⋅ln(b).

STEP 6

Our expression is now ln⁡2a3b4=ln⁡(2a3)−ln⁡(b4)\ln\frac{2a^3}{b^4} = \ln(2a^3) - \ln(b^4)lnb42a3​=ln(2a3)−ln(b4), which we've already simplified to ln⁡(2)+3⋅ln⁡(a)−4⋅ln⁡(b)\ln(2) + 3\cdot\ln(a) - 4\cdot\ln(b)ln(2)+3⋅ln(a)−4⋅ln(b).

STEP 7

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

STEP 8

The expanded form of ln⁡2a3b4\ln \frac{2 a^{3}}{b^{4}}lnb42a3​ is ln⁡2+3ln⁡a−4ln⁡b\ln 2+3 \ln a-4 \ln bln2+3lna−4lnb.

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QuestionExpand: $\ln \frac{2 a^{3}}{b^{4}}$ $\ln 2+3 \ln a-4 \ln b$ $\ln 2-3 \ln a-4 \ln b$ $\ln 2-3 \ln a+4 \ln b$

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

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

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

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 $\ln 2+3 \ln a-4 \ln b$ .