In order to solve this equation, we will need to change the base of one half to two. Use a negative exponent to rewrite this term.

\(\displaystyle \frac{1}{2}=2^{-1}\)

Rewrite the equation.

\(\displaystyle 2^{-2x} = (2^{-1})^{6x-3}\)

Since the bases are common, we can simply set the exponents equal to each other.

\(\displaystyle -2x = -(6x-3)\)

Solve for x.  Divide a negative one on both sides to eliminate the negatives.

\(\displaystyle \frac{-2x}{-1} =\frac{ -(6x-3)}{-1}\)

The equation becomes:

\(\displaystyle 2x=6x-3\)

Subtract \(\displaystyle 6x\) from both sides.

\(\displaystyle 2x-6x=6x-3-6x\)

\(\displaystyle -4x = -3\)

Divide both sides by negative four.

\(\displaystyle \frac{-4x}{-4} = \frac{-3}{-4}\)

The answer is:  \(\displaystyle \frac{3}{4}\)

When dealing with exponential equations, we want to make sure the bases are the same. This way we can set-up an equation with the exponents.

\(\displaystyle 7^{x+5}=7^{12}\) With the same base, we can now write

\(\displaystyle x+5=12\) Subtract \(\displaystyle 5\) on both sides.

\(\displaystyle x=7\)

When dealing with exponential equations, we want to make sure the bases are the same. This way we can set-up an equation with the exponents.

\(\displaystyle 12^{x+4}=12^{8}\) With the same base, we can now write

\(\displaystyle x+4=8\) Subtract \(\displaystyle 4\) on both sides.

\(\displaystyle x=4\)

When dealing with exponential equations, we want to make sure the bases are the same. This way we can set-up an equation with the exponents.

\(\displaystyle 14^{x+16}=14^{-13}\) With the same base, we can now write

\(\displaystyle x+16=-13\) Subtract \(\displaystyle 16\) on both sides.

\(\displaystyle x=-29\)

When dealing with exponential equations, we want to make sure the bases are the same. This way we can set-up an equation with the exponents.

\(\displaystyle 12^{2x+17}=12^{-13}\) With the same base, we can now write

\(\displaystyle 2x+17=-13\) Subtract \(\displaystyle 17\) on both sides.

\(\displaystyle 2x=-30\) Divide \(\displaystyle 2\) on both sides.

\(\displaystyle x=-15\)

When dealing with exponential equations, we want to make sure the bases are the same. This way we can set-up an equation with the exponents. Since the bases are now different, we need to convert so we have the same base. We do know that

\(\displaystyle 4=2^2\) therefore

\(\displaystyle 2^{3x+8}=2^2\) With the same base, we can now write

\(\displaystyle 3x+8=2\) Subtract \(\displaystyle 8\) on both sides.

\(\displaystyle 3x=-6\) Divide \(\displaystyle 3\) on both sides.

\(\displaystyle x=-2\)

When dealing with exponential equations, we want to make sure the bases are the same. This way we can set-up an equation with the exponents. Since the bases are now different, we need to convert so we have the same base. We do know that

\(\displaystyle 27=3^3\) therefore

\(\displaystyle 3^{3x-6}=3^3\) With the same base, we can now write

\(\displaystyle 3x-6=3\) Add \(\displaystyle 6\) on both sides.

\(\displaystyle 3x=9\) Divide \(\displaystyle 3\) on both sides.

\(\displaystyle x=3\)

When dealing with exponential equations, we want to make sure the bases are the same. This way we can set-up an equation with the exponents. Since the bases are now different, we need to convert so we have the same base. We do know that

\(\displaystyle 64=4^3\) therefore

\(\displaystyle 4^{2x-7}=4^3\) With the same base, we can now write

\(\displaystyle 2x-7=3\) Add \(\displaystyle 7\) on both sides.

\(\displaystyle 2x=10\) Divide \(\displaystyle 2\) on both sides.

\(\displaystyle x=5\)

When dealing with exponential equations, we want to make sure the bases are the same. This way we can set-up an equation with the exponents. Since the bases are now different, we need to convert so we have the same base. We do know that

\(\displaystyle 7^0=1\) therefore

\(\displaystyle 7^{4x+8}=7^0\) With the same base, we can now write

\(\displaystyle 4x+8=0\) Subtract \(\displaystyle 8\) on both sides.

\(\displaystyle 4x=-8\) Divide \(\displaystyle 4\) on both sides.

\(\displaystyle x=-2\)

When dealing with exponential equations, we want to make sure the bases are the same. This way we can set-up an equation with the exponents. Since the bases are now different, we need to convert so we have the same base. We do know that

\(\displaystyle 4=2^2, 8=2^3\) therefore

\(\displaystyle (2^2)^{2x+5}=(2^3)^{x}\) Apply power rule of exponents.

\(\displaystyle 2^{4x+10}=2^{3x}\) With the same base, we can now write

\(\displaystyle 4x+10=3x\) Subtract \(\displaystyle 4x\) on both sides.

\(\displaystyle 10=-x\) Divide \(\displaystyle -1\) on both sides.

\(\displaystyle -10=x\)