Question Solve the equation $8^{2x-3} = \left(\frac{1}{16}\right)^{x-2}$ for the unknown variable $x$ .

Studdy Solution

STEP 1

Assumptions
1. We are given the equation $8^{2x-3} = \left(\frac{1}{16}\right)^{x-2}$ .
2. We need to solve for the variable $x$ .

STEP 2

Recognize that both sides of the equation are written in exponential form and can be expressed as powers of the same base. The base will be 2, since $8 = 2^3$ and $16 = 2^4$ .

STEP 3

Rewrite the left side of the equation using the base 2.
$8^{2x-3} = (2^3)^{2x-3}$

STEP 4

Apply the power of a power rule $(a^b)^c = a^{bc}$ to simplify the left side of the equation.
$8^{2x-3} = 2^{3(2x-3)}$

STEP 5

Distribute the exponent on the left side.
$8^{2x-3} = 2^{6x-9}$

STEP 6

Rewrite the right side of the equation using the base 2.
$\left(\frac{1}{16}\right)^{x-2} = \left(2^{-4}\right)^{x-2}$

STEP 7

Apply the power of a power rule to simplify the right side of the equation.
$\left(\frac{1}{16}\right)^{x-2} = 2^{-4(x-2)}$

STEP 8

Distribute the exponent on the right side.
$\left(\frac{1}{16}\right)^{x-2} = 2^{-4x+8}$

STEP 9

Now that both sides of the equation are expressed as powers of 2, we can set the exponents equal to each other because the bases are the same and the exponents must be equal for the equation to hold true.
$6x - 9 = -4x + 8$

STEP 10

Add $4x$ to both sides of the equation to get all the $x$ terms on one side.
$6x - 9 + 4x = -4x + 8 + 4x$

STEP 11

Combine like terms.
$10x - 9 = 8$

STEP 12

Add 9 to both sides of the equation to isolate the term with $x$ .
$10x - 9 + 9 = 8 + 9$

STEP 13

Combine like terms.
$10x = 17$

STEP 14

Divide both sides by 10 to solve for $x$ .
$\frac{10x}{10} = \frac{17}{10}$

STEP 15

Simplify the equation to find the value of $x$ .
$x = \frac{17}{10}$
$x = 1.7$ is the solution to the equation $8^{2x-3} = \left(\frac{1}{16}\right)^{x-2}$ .

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