# solve an exponential

#### goosey00

##### Member
How do you divide using ln:

my example is x(7ln5-ln3)=-2ln5 and divide 7ln5-ln3 by both side to get x. So I have -2ln5/7ln5-ln3 and I can't seem to find the answer.

#### Jameson

##### Administrator
Staff member
How do you divide using ln:

my example is x(7ln5-ln3)=-2ln5 and divide 7ln5-ln3 by both side to get x. So I have -2ln5/7ln5-ln3 and I can't seem to find the answer.
Hi goosey00,

Try posting as much information as you can each time. This will save time by not having as many followup questions.

If you begin with $$\displaystyle x \left( 7 \ln(5)- \ln(3) \right) = -2 \ln(5)$$ then indeed one form of the answer is $$\displaystyle x = \frac{-2 \ln(5)}{\left( 7 \ln(5)- \ln(3) \right)}$$. Since this is multiple choice though, they want you to use the rules I mentioned to you yesterday about logarithms to simplify this answer further.

Another way to think of those two rules is (1) an exponent inside the natural log can be put in front of the whole expression and the reverse, (2)the natural log of a fraction is the natural log of the top minus the natural log of the bottom, and the reverse.

Remember that $$\displaystyle a \ln(x) = \ln(x^a)$$ and $$\displaystyle \ln(a/b)=\ln(a)-\ln(b)$$. Using those two rules you should be able to simplify things.

#### goosey00

##### Member
Its not multiple choice, they want me to divide and when I enter it in my calculator, I must be entering it wrong. I am entering ln(-2)5/ln(7)5-ln(3). Please don't laugh, I just am clueless.

#### MarkFL

##### Administrator
Staff member
You have found:

$\displaystyle x=-\frac{2\ln(5)}{7\ln(5)-\ln(3)}$

or

$\displaystyle x=\frac{2\ln(5)}{\ln(3)-7\ln(5)}$

You may use a calculator to get a decimal approximation for this value. IN the days of log tables. this would probably be the preferred form. If you are interested in simplifying the result, you may proceed as follows:

Divide each term by $\displaystyle \ln(5)$ to get:

$\displaystyle x=\frac{2}{\frac{\ln(3)}{\ln(5)}-7}$

This is perhaps simpler to use a calculator for which to get a decimal approximation. We may make one more change if we wish, but then most calculators will not give an approximation for this form:

Using the change of base formula, we know:

$\displaystyle \frac{\ln(3)}{\ln(5)}=\log_5(3)$ hence:

$\displaystyle x=\frac{2}{\log_5(3)-7}$

edit: Oops, sorry to post after help is given!

#### Jameson

##### Administrator
Staff member
Its not multiple choice, they want me to divide and when I enter it in my calculator, I must be entering it wrong. I am entering ln(-2)5/ln(7)5-ln(3). Please don't laugh, I just am clueless.
ln(-2) is impossible first of all The log and natural log aren't defined for 0 or negative numbers. This also depends on what kind of calculator you have. Hopefully you can enter in long expressions before hitting "equals". If so you should type something like:

(-2*ln(5))/(7*ln(5)-ln(3))

Be careful with parentheses. When you open one you must always close it somewhere. When I do it on our site's calculator (located under "MHB Widgets") I get -.317 or so.

#### goosey00

##### Member
I couldn't do the long way on the calculator. I think I can just break it down but to get your answer I was off 2 decimal places. Is that normal?

#### Jameson

##### Administrator
Staff member
I couldn't do the long way on the calculator. I think I can just break it down but to get your answer I was off 2 decimal places. Is that normal?
What do you mean off by 2 decimal places? It was plus or minus .01 from my answer? What kind of calculator do you have again?

Ti30x

#### goosey00

##### Member
I did it by doing each separately and got it as I then divided it both out. So thanks, it worked..

Staff member

#### MarkFL

##### Administrator
Staff member
My TI-89 returns ≈ -0.317