Question

please solve the equation
log (base3) 9^x =-2

the solution set is____________
type an integer or simplified fraction. Use comma to separate answers as needed)

Answer 1

are you finding the value of x??

Answer 2

i guess so since it says the solution set

Answer 3

log (base3) 9^x = x log (base3) 9
and we knowed that the value of log (base3) 9 = 2
so, we get an equation 2x = -2
solve for x

Answer 4

Okay, this would require properties of logs. Those would be: \[\log_{b}(a) + \log_{b}(c) = \log_{b}(a*c) \] \[\log_{b}(a) - \log_{b}(c) = \log_{b}(\frac{ a }{ c })\] and \[\log_{b} (a^x) = x*\log_{b} (a)\]
So first thing we would want to do is bring the x down in front of the logarithm.

Answer 5

@RadEn are you saying that x = -1
@blarghhonk8 what would we do now?

Answer 6

yup, u are right!

Answer 7

just only -1

Answer 8

exactly, yes!

Answer 9

Okay, so we now have \[x \log_{3} (9) = -2\]
Well, what does a logarithm tell us? We typically use logarithms when we have an equation like the following: \[b^x=a \] Here we know b and a, but don't know x. However, logarithms are used to find x because \[\log_{b} (a) =x\] So we know from the equation given that \[3^{-2} = 9x\]

Answer 10

okay now what? solve for x?

Answer 11

Yep. (remember that \[x ^{-a} = \frac{ 1 }{ x^{a} }\])

Answer 12

so would it be 1/81 but that RadEn says it is -1

Answer 13

One moment, let me work it out using a calculator.

Answer 14

Oh! I'm sorry. I forgot you that's not quite how it would work out. When we bring the x down in front we'd have to use a calculator to calculate \[\frac{-2}{\frac{log(9)}{log(3)}}\] To find x since x is technically "attached" to the log(base3) (9). So instead what we would do is actually rewrite \[\log_{b} a^{x} = c\] as \[b ^{c} = a ^{x}\] and solve from there. So we would be solving \[3^{-2}=9^{x}\]

Answer 15

(and this is of course if you just want/have to solve it without a calculator. )