Need help with solving equations involving logarithms

[green elbow]

The correct answer is E, and I'm struggling to see how I can get that answer. I got to (1-m)/m = 3^(2x) but I'm not sure where to go from there. Any help would be greatly appreciated, thank you.

[mild ibex]

yes, multiplying both sides by m is a good step. at this point, you can treat 3^2x as just some number (you can replace it with some variable if you prefer)

[green elbow]

that's what I got, but that answer is different from option E

[mild ibex]

double check your work from the 2nd to the 3rd line

[green elbow]

from diving both sides with 1 I got -m = 3^(2x) * m, then that third line is me dividing both sides by -1. i'm confused?

[mild ibex]

the left hand side goes from 1-m to -m after dividing by -1?

[green elbow]

ohhh i see

[mild ibex]

the higher level issue here is that dividing by -1 really doesn't help solve for m

[green elbow]

divding 1-m by 1 is just 1-m still right?

[mild ibex]

yep

I recommend taking a step back.
if you had a problem like
1-m = 55*m
how would you solve for m?

[green elbow]

divide by 55?

[mild ibex]

well, when we're "solving" for m, we're really trying to move all the m values to one place
if you divide by 55, there's still an m on the left and right hand sides, so it isnt any better.

you could instead add m to both sides, and that removes the m on the left

or similarly useful, you could subtract each side by 55m

[green elbow]

so i would do this next? I'm just not understanding what I'm supposed to be doing on the left, like how I can get m by itself

[mild ibex]

yeah, that's why it was better to have multiplied both sides by m (per your original question), which this newest work just undid

a fraction where m is in multiple places is complicated, so we want to stretch it out, bringing us to/returning us to:
1 - m = 3^2x * m

we see some terms have m as a factor, and some don't, so the next best step is to consolidate them, so we can factor m off
1 - m + m = 3^2x * m + m

1 = 3^2x * m + m

1 = (3^2x + 1) * m (factored off m)

[green elbow]

ohhhh ok I understand now, I didn't even think about factoring the right side

thanks for the help

[mild ibex]

🥳

[green elbow]

.solved