Please show steps?

Question

solve: 5log (subscript 10^x) + log (subscript 10^5) = 1 + 2log (subscript 10^4)

factorise: a - 2ab - 2bc + c

value of t so that: 9x^2 + 16x + t is a perfect square

Answer 1

5 log x + log 5 = 1 + 2 log 4
5 log x = 1 + 2 log 4 - log 5
5 log x = log 10 + 2 log 4 - log 5
5 log x = log 10 + log (4^2) - log 5
5 log x = log (10*16/5)
5 log x = log 32
log x^5 = log 32
x^5 = 32
x = 2

a - 2ab - 2bc + c
= a(1-2b) + c(1-2b)
= (a+c)(1-2b)

9x^2 + 16x + t = (ax+b)^2 for some a,b
(ax+b)^2 = a^2 x^2 + 2abx + b^2 and so a=3 and so b=8/3 and therefore t=(8/3)^2

Answer 2

For the first question, I would suggest that only the ten should be subscript, what you have indicated as a power is actually just not subscript. Also, if it's just ten, the subscript isn't strictly neccessary.
This leads me to believe that you don't know what you're talking about and are just posting you homework on Yahoo Answes to get out of doing it yourself... but I'll help you anyway.

First question: remeber your log laws. Remember that log10=1.
This gives you 5x^5=160. You can work it out from there.

Second question: what gimb said.

Third question: Do you know what a perfect square is?... I didn't think so. A perfect square is a quadratic function that can be expressed as (ax + b)^2.
For this question, because the first coefficient is 9, we know that a must be the square root of 9, 3. So (3x +b)^2 = 9x^2 +16x + t.
If we expand the first expression we get (3x + b)^2 = 9x^2 + 3bx + 3bx + b^2. So 3b + 3b = 16. You can take it from there.

Answer 3

a - 2ab - 2bc + c = a(1-2b) + c(1-2b) = (a+c)(1-2b)
9x^2 + 16x + t = (3x + a )^2 = 9x^2 + 6ax + a^2
6a = 16 => a = 8/3 => t=a^2 = 64/9

thus t = 64/9.

all logs are 10 based i assume
5log (x) + log (5) = 1 + 2log (4)
=> x = 10^(1/5*(1+log(4) - log(5)))

Answer 4

5 log_10 x + log_10 5 = 1 + 2 log_10 4

Use these identities:
log_b b = 1
n log_b x = log_b x^n
log_b x + log_b y = log_b xy

So,
5 log_10 x + log_10 5 = 1 + 2 log_10 4

log_10 x^5 + log_10 5 = log_10 10 + log_10 4^2
log_10 5(x^5) = log_10 (10 · 4^2)
5x^5 = 160
x^5 = 32
x = 2

^_^



a - 2ab - 2bc + c

Regroup
a + c - 2ab - 2bc

Factor common monomial
(a + c) - 2b(a + c)
(a + c)(1 - 2b)

^_^



9x² + 16x + t is a perfect square, so the discriminant (b² - 4ac) is equal to zero.
a = 9
b = 16
c = t
so
16² - 4(9)(t) = 0
256 - 36t = 0
36t = 256
9t = 64
t = 64/9

^_^

Answer 5

Your third one:

= (3x)^2 + 2 * 3x * 8/3 + t

=> t = (8/3)^2

For the first one, dont understand what u meant by (subscript 10^x)

Answer 6

Your second question:

Factorise a - 2ab - 2bc + c

a(1 - 2b) - c(2b - 1)