solve the log?

Question

logx^2 + logx^3 / log(100x) = 3

all logs in base 10. can someone show me how to solve this step by step?

Answer 1

Assuming question is meant to read as:-
( log x ² + log x ³ ) / log (100 x) = 3
log x ² + log x ³ = log (100 x) ³
log x^5 = log (100 x) ³
x^5 = 100 ³ x ³
x ² = 100 ³
x = 100^(3/2)
x = 1000

Answer 2

There are two solutions
x = 10 & x = 1/1000

logx^2 + logx^3 / log(100x) = 3
2logx + 3logx / log(100x) = 3
2logx + 3logx / (log(100) + logx) = 3
2logx + 3logx / (2+ logx) = 3
2(2+ logx)logx + 3logx = 3(2+ logx)
2(logx)² + 4(logx) - 6 = 0
(logx)² + 2(logx) - 3 = 0
this is a quadratic in terms of logx which can be factored as
(logx - 1)(logx + 3) = 0
gives
x = 10^1 = 10
&
x = 10^-3 = 1/1000
as the two solutions

Edit
Regardless of the thumbs down, this is correct
just substitute the two answers in to your original equation with your calculator to confirm this

,.,.

Answer 3

Multiply tro` by log 100x:

log x^2. log (100 x) +log x^3 = 3 log(100x),so

2logx.log(100x) + 3logx = 3log(100x)

log 100x = (2 + log x), so

2logx(2 + logx) + 3logx = 3(2 + logx)

4 logx +2 logx.logx + 3 log x = 6 + 3logx

4logx + 2logx.logx = 6

2logxlogx + 4logx - 6 = 0

(2logx - 2)(logx + 3) = 0

2logx - 2 = 0, or log x + 3 = 0, so

2 logx = 2, or logx = -3, so

x^2 = 100, or x = 10^-3, so

x = +/- 10, or x = 1/1000

I think the only acceptable answer is x = 10, which is correct when substituted into the original equation.

Hope this helps, Twiggy.

Answer 4

I assume you mean
[ logx^2 + logx^3 ] / log(100x) = 3
then we have
2*logx + 3* logx = 3* log(100x) = 3[ logx + log 100 ]
and so
2logx = 6 or x = 10^3

Here we use log x^m = mlogx , log (a*x) = loga + logx etc

Answer 5

whilst accessible, use residences... for this reason it quite is accessible. the valuables you need to use is: The log of ability of the backside is the exponent... this is... in base 4 log(4^7) = 7 First you write (a million/sixteen) as ability of four.... (a million/sixteen)^3 = (4^-2) And now practice the valuables... (base 4) [log4^(-2)]^3 = [-2]^3 = -8 ok!

Answer 6

logx^2 + logx^3 /(log100 + logx) =3
logx^2 + logx^3 /(log10^2 +logx) =3
2logx + 3logx/(2 + logx) =3
2logx(2 + logx) + 3logx= 3(2 + logx)
2logx(2+ logx) =3*2
logx(2 + logx) - 3 =0
let logx=t
t^2 + 2t -3=0
t = -3, 1
logx = -3 and logx=1
x=10^(-3) and x=10

Answer 7

Put the other side into the same base
log(x^2) + log(x^3) / log(100x) = log10^3

Remember your laws. . .
log [(x^2 * x^3) / 100x] = log(10^3)

Since all are in same bases now, you can put them into exponent form
(x^2 * x^3) / 100x = 10^3

Simplify:
(x^2 * x^3) = (1000)(100x) - simplified 10^3
x^6 = 100 000x - isolate for 100 000, so divide both sides by x
x^5 = 100 000 - put into square root
sqrt(5) x = sqrt(5) 100 000
x = 10

Answer 8

here's how it's solved:
logx^2+logx^3/Log100x=2logx+logx^3/100x=2logx+3logx-log100x=2logx+3logx-(log100+logx)=5logx-2-logx=4logx-2=3--->4logx=5--->logx=5/4--->x=10^5/4