Question

Please help:

Solve using logarithms. Round the solution to the nearest hundredth. Show your work.
(you will need to use a calculator at some point)

Answer 1

To solve this problem we are going to use the power rule of logarithms:
`ln(a^(x) )=xln(a)`. Or in words: the logarithm of a power is the power times the logarithm of the base.
So, the first thing we are going to do is take natural logarithm,
`ln`, to both sides of our equation:

`ln(5^(x+2) )=ln(9^(2x+3) )`
Now, by the power rule of logarithms:

`(x+2)ln(5)=(2x+3)ln(9)`

`((x+2)ln(5))/(2x+3) =ln(9)`

`(x+2)/(2x+3)= (ln(9))/(ln(5))`

`(x+2)/(2x+3) =1.365`

`x+2=1.365(2x+3)`

`x+2=2.73x+4.095`

`-1.73x=2.095`

`x=- (2.095)/(1.73)`

`x=-1.211`

Using our calculator we can conclude that
`x` is approximately -1.211