If 3^{x+4} = 9^x, what is the value of x? — SAT

If $3^{x+4} = 9^x$ , what is the value of $x$ ?

To solve the equation

3^{x+4} = 9^x

, we rewrite the base 9 as

3^2

, which gives

3^{x+4} = (3^2)^x

. Applying the exponent rule

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

, we get

3^{x+4} = 3^{2x}

. Since the bases are now the same, we can set the exponents equal to each other:

x+4 = 2x

. Subtracting

x

from both sides yields the solution

4

Express both sides of the equation with a common base.

Since

9 = 3^2

, the equation

3^{x+4} = 9^x

can be rewritten as

3^{x+4} = (3^2)^x

. Using the power of a power rule, this becomes

3^{x+4} = 3^{2x}

To solve an exponential equation, it is helpful to have the same base on both sides so that the exponents can be equated.

Set the exponents equal to each other.

x + 4 = 2x

Since the bases are equal and positive (and not equal to 1), their exponents must be equal.

Solve the linear equation for

x

4 = x

(or

x = 4

)

Subtract

x

from both sides of the equation to isolate the variable.

Solving exponential equations by expressing terms with a common base and equating their exponents.

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