If 3^{x - 2} = 81, what is the value of x? — SAT

If $3^{x - 2} = 81$ , what is the value of $x$ ?

To solve the equation

3^{x - 2} = 81

, we first express the number 81 as a power of 3, which is

3^4

. This gives us

3^{x - 2} = 3^4

. Since the bases are equal, we can set their exponents equal to each other, resulting in the equation

x - 2 = 4

. Solving for

x

by adding 2 to both sides gives the correct value of 6.

Express both sides of the equation with a common base of 3.

3^{x - 2} = 3^4

Since 81 is equal to

3 \times 3 \times 3 \times 3

, it can be written as

3^4

Set the exponents equal to each other because the bases are now the same.

x - 2 = 4

b^y = b^z

for a positive base

b \neq 1

, then

y = z

Solve the linear equation for

x

by adding 2 to both sides of the equation.

x = 6

Adding 2 to both sides isolates the variable

x

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

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