Question

HELP!!! someone? Explain this to me please!

Suppose a house that costs $270,000 appreciates by 5% each year.
In about how many years will the house be worth $350,000? Use the equation 350 = (270)(1.05)x and round the value of x to the nearest year.
5
4
6
7

Answer 1

So we have an exponential equation:\[\Large\rm A=P(b)^x\]Where P is the principle. (Starting amount).
We assume the house will be worth 350 (thousand) at some point, so we let our A be that amount. b represents the growth rate.\[\Large\rm 350=270(1.05)^x\]To solve for x we'll need to use logarithms.

Answer 2

Let's start by dividing each side by 270,\[\Large\rm \frac{350}{270}=1.05^x\]

Answer 3

So itll be 1.29= 1.05^x

Answer 4

1.30 is probably a better way to round that.

Answer 5

True

Answer 6

Next, we'll take the log of each side (log base 10 is fine),\[\Large\rm \log(1.30)=\log(1.05^x)\]

Answer 7

0.1139=log(1.05^x)

Answer 8

The log is helpful for getting variables out of the exponent position.

Answer 9

Hmm but my calculator doesnt register x as a exponent

Answer 10

Here is the log rule that will help us:\[\Large\rm \log(a^\color{orangered}{x})=\color{orangered}{x} \log(a)\]

Answer 11

We're not up to the calculator step yet ;)

Answer 12

Oh okay! so it would look like 0.1139=xlog(1.05)

Answer 13

Mmm good good good.
Just a few more steps to finish it up from there.

Answer 14

Okays :D

Answer 15

It seems like you prefer to carry the decimals around, rather than the logs.
So from here, let's get our decimal approximation for log(1.05)

Answer 16

its 0.0211

Answer 17

0.1139=0.0211x

ok good.
last step? :d any ideas?

Answer 18

divide 0.1139 by 0.0211= 5.39 = 5.4

Answer 19

so 5?

Answer 20

yay good job \c:/

Answer 21

Yay! thank you so much! :)

Answer 22

np :3