Help with algebra 2 Questions?

Question

I need help with a couple problems:

ln e^ln x - ln(x-3) = ln 2

2^x+3=5^x

2e^5x+2=8

My problem is that I need exact answers. Can you help? Thanks.

Answer 1

for the first one:

ln e^ln x - ln(x-3) = ln 2

assuming you meant ln(e^[ln(x)]) - ln(x-3) = ln(2),

then that is equal to ln(x) - ln(x-3) = ln(2)

then ln(x/[x-3]) = ln(2)

then x/(x-3) = 2

then x = 2x - 6

then x = 6

second one:

assuming you meant 2^(x+3) = 5^x

which equals 2^x times 2^3 = 5^x

then take the x root of both sides resulting in 2 times the x root of 8 equals 5

then divide both sides by 2

this leaves you with something that can be written in log form, which is log 8 (base 5/2) = x

third one:

assuming you meant 2 times e^(5x+2) = 8

then, divide both sides by 2, e^(5x+2) = 4

then, take the ln of both sides, 5x+2 = ln(4)

then, 5x = ln(4) -2

x = [ln(4)-2]/5

added question #1:

They cancel out resulting in just 8 as your answer

added question # 2:

just multiply the bottom equation by -9 and then solve for y, then plug it into the first equation to retrieve x, then test. This should yield:

y = sqrt(45/77) x = sqrt(288/77)