Question

if 4^x - 4^x-1 = 24 find the value of x

Answer 1

4^x-4^x-1=24

4^x-4^x × 4^-1=24

4^x (1-4^-1)=24

4^x (1-1÷4)=24

(taking LCM)

4^x (4-1÷4)=24

4^x (3÷4)=24

4^x=24×4÷3

(on solving 24÷3 =8)

4^x=8×4÷1

4^x=32

(2^2)^x=2^5

2^2x=2^5

Sice the bases are same we equate the powers,therefore

2x=5

x=5÷2

So the answer is 5÷2 or 2.5

Answer 2

Given:

${4}^{x} - {4}^{(x - 1)} = 24$

To find:

The value of x.

Solution:

To answer this question, we should know some of the properties of exponents and powers.

${x}^{a} \times {x}^{b} = {x}^{(a + b)}$

${x}^{ - 1} = \frac{1}{x}$

Now,

As given, we have,

${4}^{x} - {4}^{(x - 1)} = 24$

${4}^{x} - {4}^{x } . {4}^{ - 1} = 24$

After taking 4 with power x common, we have,

${4}^{x} \: (1 - {4}^{ - 1} ) = 24$

${4}^{x} \: (1 - \frac{1}{4} ) = 24$

${4}^{x} \: ( \frac{3}{4} ) = 24$

(On taking 4 as LCM)

${4}^{x} = \frac{24 \times 4}{3}$

${4}^{x} = 8 \times 4$

${4}^{x} = 32$

${2}^{2 x } = {2}^{5}$

$(as \: 4 = {2}^{2} and \: {x}^{ {2}^{(a)} } = {x}^{2a})$

$(32 = {2}^{5} )$

As the base is the same, so, on comparing the powers, we have,

$2x = 5$

$x = \frac{5}{2}$

$x = 2.5$

Hence, the value of x is 5/2 or 2.5.