Math  /  Algebra

QuestionExpand each expression. ln(2x)4\ln (2 x)^{4} 4ln2+4lnx4 \ln 2+4 \ln x 4ln2+lnx4 \ln 2+\ln x 8lnx8 \ln x

Studdy Solution

STEP 1

What is this asking? We need to rewrite ln(2x)4\ln (2x)^4 in a simpler form using logarithm rules. Watch out! Don't forget the logarithm power and product rules!

STEP 2

1. Apply the power rule
2. Apply the product rule

STEP 3

Alright, let's **kick things off** with our expression ln(2x)4\ln (2x)^4.
The power rule for logarithms says ln(ab)=bln(a)\ln(a^b) = b \cdot \ln(a).
This means we can bring that **powerful** exponent of 4\bf{4} down in front of the logarithm.

STEP 4

So, ln(2x)4\ln (2x)^4 becomes 4ln(2x)4 \cdot \ln (2x).
See? **Easy peasy**!

STEP 5

Now, we've got 4ln(2x)4 \cdot \ln (2x).
Remember the product rule?
It says ln(ab)=ln(a)+ln(b)\ln(a \cdot b) = \ln(a) + \ln(b).
This lets us **break apart** that ln(2x)\ln(2x) into two separate logarithms.

STEP 6

Applying the product rule, we get 4(ln2+lnx)4 \cdot (\ln 2 + \ln x).
We're almost there!

STEP 7

Finally, **distribute** that 4\bf{4} to both terms inside the parentheses: 4ln2+4lnx4 \cdot \ln 2 + 4 \cdot \ln x.
And there you have it!

STEP 8

Our **final answer** is 4ln2+4lnx4 \cdot \ln 2 + 4 \cdot \ln x.

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