We have, log3 x = 0

So, 30 = x

1 = x

Hence, x = 1

We have,

log10 x = -2

10-2 = x [As loga c = b ? ab = c]

x = 10-2

x = 1/102

x = 1/100

Hence, x = 0.01

We know that,

loga c = b ? ab = c

(i) log8 0.125 = -1

8-1 = 0.125

10-3 = 0.001

log10 0.001 = -3 

Given equations are 8x + 5y = 9 and 3x + 2y = 4

On comparing with a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0, we have

a1 = 8, b1 = 5, c1 = -9 and a2 = 3, b2 = 2, c2 = -4

Now, x = (b1c2 – b2c1)/(a1b2 – a2b1) and y = (c1a2 – c2a1)/(a1b2 – a2b1)

x = [5 × (-4) – 2 × (-9)]/[8 × 2 – 3 × 5] and y = [ -9 × 3 – (-4) × 8]/[8 × 2 – 3 × 5]

x = (-20 + 18)/(16 – 15) and y = (-27 + 32)/(16 – 15)

x = -2/1 and y = 5/1

Therefore, x = -2 and y = 5

Given equations are v2x – v3y = 0 and v5x + v2y = 0

On comparing with a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0, we have

a1 = v2, b1 = -v3, c1 = 0 and a2 = v5, b2 = v2, c2 = 0

Now, x = (b1c2 – b2c1)/(a1b2 – a2b1) and y = (c1a2 – c2a1)/(a1b2 – a2b1)

x = [(-v3) × 0 – v2 × 0]/[v2 × v2 – v5 × (-v3)] and y = [0 × v5 – 0 × v2]/[v2 × v2 – v5 × (-v3)]

x = [0/(2 + v15)] and y = [0/(2 + v15)]

Therefore, x = 0 and y = 0

Given,

The value of expression mx – ny is 3 when x = 5 and y = 6.

? 5m – 6n = 3 … (i)

And,

The value of expression mx – ny is 8 when x = 6 and y = 5

? 6m – 5n = 8 … (ii)

Solving for m and n:

Performing (i)×6 – (ii)×5, we get

30m – 36n = 18

30m – 25n = 40

(-) (+) (-)

____________

-11n = -22

So,

n = 22/11

n = 2

Now, on substituting n = 2 in equation (i), we get

5m – 6(2) = 3

5m = 15

m = 3

? The solution is m = 3 and n = 2.

Given, x = 2 and y = 1 is the solution of the equation 2x + 3y = m

Then,

2(2) + 3(1) = m

4 + 3 = m

? m = 7

Hence, if x = 2 and y = 1 is the solution of the equation 2x + 3y = m, then the value of m = 7.