How do you solve 4ln(5x)+5=2?

2 Answers
Jul 6, 2017

x = e^(-3/4)/5

Explanation:

In order to solve this, we will begin with removing 5 from both sides:
4ln(5x) + 5 - 5 = 2 - 5.

Thus, we will get 4ln(5x) = -3.

Then we will divide both sides by 4 to get ln(5x) = -3/4.

Then, to get x alone, we need to recognize the relationship of e and ln. If we rewrite both sides as e^(ln(5x)) = e^(-3/4) we can perform our next step.

Remember that ln and e are inverses of each other and that ln(e) = 1. Similarly, e^ln(x) = x. If you look at e^ln(5x), you will recognize that we will get 5x due to ln and e undoing each other.

Thus, continuing: 5x = e^(-3/4).

Divide both sides by 5 and we will get x = e^(-3/4)/5.

If you plug this into the equation for x, you will get 4ln(5(e^(-3/4)/5)) + 5 = 2.

Jul 6, 2017

x = e^(-3/4)/5

Explanation:

Isolate the logarithm and then cancel it out by raising everything to a common base.

First, subtract 5 from both sides.

4ln(5x)+5-5=2-5

4ln(5x)+cancel5-cancel5=2-5

4ln(5x) = -3

Next, divide both sides by 4.

4ln(5x) div 4 = -3 div 4

cancel4ln(5x) div cancel4 = -3 div 4

ln(5x) = -3/4

Next, raise e to the power of both sides. This will cancel out the natural logarithm.

e^ln(5x) = e^(-3/4)

cancele^(cancel"ln"(5x))=e^(-3/4)

5x = e^(-3/4)

And finally divide by 5.

5x div 5 = e^(-3/4)div5

cancel5x div cancel5 = e^(-3/4)/5

x = e^(-3/4)/5

Final Answer