Answer
405k+ views
Hint: Convert the terms involving the variable x, in powers of the same number.
It can be observed that $25={{5}^{2}}$ .
The following properties of indices (powers) are useful.
(i) ${{a}^{x}}\times {{a}^{y}}={{a}^{x+y}}$
(ii) $\dfrac{{{a}^{x}}}{{{a}^{y}}}={{a}^{x-y}}$
(iii) ${{({{a}^{x}})}^{y}}={{a}^{xy}}$
Complete step-by-step answer:
Let us start simplifying by converting all the terms which have the variable x, into powers of 5.
${{5}^{2x-1}}-{{25}^{x-1}}=2500$
⇒ ${{5}^{2x-1}}-{{({{5}^{2}})}^{x-1}}=2500$
Using the property that ${{({{a}^{x}})}^{y}}={{a}^{xy}}$ , the above expression equals:
⇒ ${{5}^{2x-1}}-{{5}^{2}}^{x-2}=2500$
And, using the property that $\dfrac{{{a}^{x}}}{{{a}^{y}}}={{a}^{x-y}}$ , the above can be written as:
⇒ $\dfrac{{{5}^{2x}}}{{{5}^{1}}}-\dfrac{{{5}^{2x}}}{{{5}^{2}}}=2500$
Let us substitute ${{5}^{2x}}=y$ to reduce writing work and a simpler look of the equation:
⇒ $\dfrac{y}{5}-\dfrac{y}{25}=2500$
Multiply both the numerator and denominator of $\dfrac{y}{5}$ by 5, to get same denominators on the LHS:
⇒ $\dfrac{5y}{25}-\dfrac{y}{25}=2500$
Multiplying both sides of the equation by 25, so that we are left with only the variable on the LHS:
⇒ 5y - y = 25 × 2500
⇒ 4 y = 25 × 25 × 100
Dividing both sides of the equation by 4:
⇒ y = 25 × 25 × 25
⇒ $y={{5}^{2}}\times {{5}^{2}}\times {{5}^{2}}$
Using the property ${{a}^{x}}\times {{a}^{y}}={{a}^{x+y}}$ , we get:
⇒ $y={{5}^{2+2+2}}$
⇒ $y={{5}^{6}}$ .
In order to solve for x, let us back-substitute ${{5}^{2x}}=y$ in the above value of y.
⇒ ${{5}^{2x}}={{5}^{6}}$
Since the bases are the same, the powers must also be the same.
∴ 2x = 6
⇒ x = 3.
Hence, the value of x is 3.
Note: If ${{a}^{x}}{{b}^{y}}={{a}^{p}}{{b}^{q}}$ , then it is not necessarily true that x = p and y = q.
But if ${{a}^{x}}={{a}^{p}}$ , then it is always true that x = p.
It can be observed that $25={{5}^{2}}$ .
The following properties of indices (powers) are useful.
(i) ${{a}^{x}}\times {{a}^{y}}={{a}^{x+y}}$
(ii) $\dfrac{{{a}^{x}}}{{{a}^{y}}}={{a}^{x-y}}$
(iii) ${{({{a}^{x}})}^{y}}={{a}^{xy}}$
Complete step-by-step answer:
Let us start simplifying by converting all the terms which have the variable x, into powers of 5.
${{5}^{2x-1}}-{{25}^{x-1}}=2500$
⇒ ${{5}^{2x-1}}-{{({{5}^{2}})}^{x-1}}=2500$
Using the property that ${{({{a}^{x}})}^{y}}={{a}^{xy}}$ , the above expression equals:
⇒ ${{5}^{2x-1}}-{{5}^{2}}^{x-2}=2500$
And, using the property that $\dfrac{{{a}^{x}}}{{{a}^{y}}}={{a}^{x-y}}$ , the above can be written as:
⇒ $\dfrac{{{5}^{2x}}}{{{5}^{1}}}-\dfrac{{{5}^{2x}}}{{{5}^{2}}}=2500$
Let us substitute ${{5}^{2x}}=y$ to reduce writing work and a simpler look of the equation:
⇒ $\dfrac{y}{5}-\dfrac{y}{25}=2500$
Multiply both the numerator and denominator of $\dfrac{y}{5}$ by 5, to get same denominators on the LHS:
⇒ $\dfrac{5y}{25}-\dfrac{y}{25}=2500$
Multiplying both sides of the equation by 25, so that we are left with only the variable on the LHS:
⇒ 5y - y = 25 × 2500
⇒ 4 y = 25 × 25 × 100
Dividing both sides of the equation by 4:
⇒ y = 25 × 25 × 25
⇒ $y={{5}^{2}}\times {{5}^{2}}\times {{5}^{2}}$
Using the property ${{a}^{x}}\times {{a}^{y}}={{a}^{x+y}}$ , we get:
⇒ $y={{5}^{2+2+2}}$
⇒ $y={{5}^{6}}$ .
In order to solve for x, let us back-substitute ${{5}^{2x}}=y$ in the above value of y.
⇒ ${{5}^{2x}}={{5}^{6}}$
Since the bases are the same, the powers must also be the same.
∴ 2x = 6
⇒ x = 3.
Hence, the value of x is 3.
Note: If ${{a}^{x}}{{b}^{y}}={{a}^{p}}{{b}^{q}}$ , then it is not necessarily true that x = p and y = q.
But if ${{a}^{x}}={{a}^{p}}$ , then it is always true that x = p.
Recently Updated Pages
How many sigma and pi bonds are present in HCequiv class 11 chemistry CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Why Are Noble Gases NonReactive class 11 chemistry CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let X and Y be the sets of all positive divisors of class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let x and y be 2 real numbers which satisfy the equations class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let x 4log 2sqrt 9k 1 + 7 and y dfrac132log 2sqrt5 class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let x22ax+b20 and x22bx+a20 be two equations Then the class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Trending doubts
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
At which age domestication of animals started A Neolithic class 11 social science CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Which are the Top 10 Largest Countries of the World?
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Give 10 examples for herbs , shrubs , climbers , creepers
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Difference Between Plant Cell and Animal Cell
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Write a letter to the principal requesting him to grant class 10 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Change the following sentences into negative and interrogative class 10 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Fill in the blanks A 1 lakh ten thousand B 1 million class 9 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)