Question

if 25 to the power x - 1 + 100 = 5 to the power 2 x minus 1 find the value of x

Answer 1

Answer:

2

Step-by-step explanation:

Given,

$25^{x-1} + 100 = 5^{2x-1}$

To find the value of x

we know that, 25 = $5^{2}$

So, $25^{x-1}$ = $(5^{2})^{x-1}$

= $5^{2x - 2}$

(since, $(a^{m})^{n}$ = $a^{mn}$)

So now, we have

$25^{x-1} + 100 = 5^{2x-1}$

=$5^{2x - 2} + 100 = 5^{2x-1}$

=> 100 = $5^{2x-1} - 5^{2x - 2}$

Now, $5^{2x-1} = \frac{5^{2x}}{5}$

So, 100 = $\frac{5^{2x}}{5} -\frac{5^{2x}}{25}$

Take $5^{2x}$ out as common

=> 100 = $5^{2x}(\frac{1}{5} -\frac{1}{25})$

$=> 100 = 5^{2x}(\frac{5}{25} -\frac{1}{25})\\\\=> 100 = 5^{2x}(\frac{4}{25})\\\\=> 5^{2x} =100 \times \frac{25}{4}$

$=> 5^{2x} = 625\\\\=> 5^{2x} = 5^{4}$

Now comparing the powers, since the bases are same, we get

2x = 4

Hence, x = 4/2

x = 2

Answer 2

Q1) $25^{(x-1)} \: + \: 100 \: = \: 5^{(2x-1)}$

Revise formula :

$m^{a-b} \: = \: \dfrac{m^{a}}{m^{b}}$

Answer :

$5^{2(x-1)} \: + \: 2^{2}.5^{2}\: =\: 5^{(2x-1)}$

$5^{2(x-1)} \: -\: 5^{(2x-1)}\: =\: - 2^{2}.5^{2}$

$\dfrac{5^{2x}}{5^{2}}\: - \: \dfrac{5^{2x}}{5} = \: -2^{2}.5^{2}$

$5^{2x}\Big[\dfrac{1-5}{25}\Big]\: = \: -2^{2}.5^{2}$

$5^{2x}\Big[\dfrac{-4}{25}\Big] \: = \: -2^{2}.5^{2}$

$5^{2x} \: = \dfrac{2^{2}.5^{4}}{4}$

$5^{2x} \: = \dfrac{2^{2}.5^{4}}{2^{2}}$

$5^{2x}\: = \: 5^{4}$

now taking power

2x = 4

$x\: = \: \dfrac{4}{2}$

$\bold{\boxed{x\: = \: 2}}$