If 4^x + 4^x + 4^x + 4^x = 2^{10}, what is the value of x? — GMAT

If $4^x + 4^x + 4^x + 4^x = 2^{10}$ , what is the value of $x$ ?

The value of

x

4

Combining the four identical terms on the left side gives

4 \cdot 4^x = 4^{1+x}

. Converting

4

to base

2

yields

(2^2)^{x+1} = 2^{2x+2}

. Setting this equal to the right side

2^{10}

gives the linear equation

2x + 2 = 10

, which solves to

x = 4

Combine the repeated addition on the left side of the equation.

4^x + 4^x + 4^x + 4^x = 4 \cdot 4^x = 4^{x+1}

Adding four identical terms is equivalent to multiplying the term by 4.

Convert all bases to 2 so that both sides can be compared.

4^{x+1} = (2^2)^{x+1} = 2^{2(x+1)} = 2^{2x+2}

Since

4 = 2^2

, applying the power of a power rule

(a^m)^n = a^{mn}

expresses the left side with base 2.

Equate the exponents and solve for

x

2x + 2 = 10 \implies 2x = 8 \implies x = 4

b^u = b^v

for a positive base

b \neq 1

, then

u = v

Combining like exponential terms by factoring and applying base conversion rules

a^{m} \cdot a^{n} = a^{m+n}

and

(a^m)^n = a^{mn}

Question