# Logarithmic word problem

#### otownsend

Hi,

I hope someone can help me with the following question:

Can someone please let me know whether my solution is correct? I said that in about in year 2015, the amount pests would be the same. The reason why I want to know whether I'm right is because my solution does not match up to the textbook's.

I would really appreciate help!

#### skeeter

MHF Helper
I get 15.93 years from 1997 ... the initial population of the Washington forest is 1.95 million, not 1.9 million

#### HallsofIvy

MHF Helper
I get a slightly different answer. I used the fact that $$\displaystyle \frac{0.97^x}{0.96^x}= \left(\frac{0.97}{0.96}\right)^x= 1.0104^x$$ so that $$\displaystyle 1.0104^x= 1.2105$$. Now take the logarithm of both sides: $$\displaystyle x log(1.0104)= log(1.2105)$$ so $$\displaystyle 0.00500x= 0.08296$$ and the $$\displaystyle x= \frac{0.08296}{0.00500}= 16.6, so the answer is 1996+ 17= 2013 rather than 2015. What was the answer in your text? It might be just a matter of round off error.$$

#### otownsend

I get a slightly different answer. I used the fact that $$\displaystyle \frac{0.97^x}{0.96^x}= \left(\frac{0.97}{0.96}\right)^x= 1.0104^x$$ so that $$\displaystyle 1.0104^x= 1.2105$$. Now take the logarithm of both sides: $$\displaystyle x log(1.0104)= log(1.2105)$$ so $$\displaystyle 0.00500x= 0.08296$$ and the $$\displaystyle x= \frac{0.08296}{0.00500}= 16.6, so the answer is 1996+ 17= 2013 rather than 2015. What was the answer in your text? It might be just a matter of round off error.$$
$$\displaystyle Yeah you're right, it was just a round off error. Thanks for your help!$$

#### DenisB

230 * (1 - .04)^n = 195 * (1 - .03)^n

Solve for n (you should get 15.93005335275....)

Remember that a^n / b^n = (a/b)^n

#### skeeter

MHF Helper
$2.3(0.96)^x = 1.95(0.97)^x$

$\dfrac{2.3}{1.95} = \dfrac{0.97^x}{0.96^x} = \left(\dfrac{0.97}{0.96}\right)^x$

$\log\left(\dfrac{2.3}{1.95}\right) = x\log\left(\dfrac{0.97}{0.96}\right)$

$\dfrac{\log \left( \frac{2.3}{1.95} \right)}{\log \left( \frac{0.97}{0.96} \right)} = x$

$15.93 = x$

#### otownsend

Thanks, I got it!