Question Solve for $x$ in the equations $8^{x}=4$ and $16^{x}=8$ .

Studdy Solution

STEP 1

1. The equations $8^{x}=4$ and $16^{x}=8$ are exponential equations where the base and the result are both powers of 2.
2. The value of $x$ can be found by expressing both sides of the equation with the same base and then equating the exponents.
3. The laws of exponents will be used to solve the equations.

STEP 2

1. Solve the equation $8^{x}=4$ for $x$ .
2. Solve the equation $16^{x}=8$ for $x$ .

STEP 3

Express both sides of the equation $8^{x}=4$ with the same base.
Since $8 = 2^3$ and $4 = 2^2$ , we can rewrite the equation as:
$(2^3)^x = 2^2$

STEP 4

Apply the power of a power rule to the left side of the equation.
$2^{3x} = 2^2$

STEP 5

Since the bases are now the same, we can equate the exponents.
$3x = 2$

STEP 6

Divide both sides of the equation by 3 to solve for $x$ .
$x = \frac{2}{3}$

STEP 7

Express both sides of the equation $16^{x}=8$ with the same base.
Since $16 = 2^4$ and $8 = 2^3$ , we can rewrite the equation as:
$(2^4)^x = 2^3$

STEP 8

Apply the power of a power rule to the left side of the equation.
$2^{4x} = 2^3$

STEP 9

Since the bases are the same, equate the exponents.
$4x = 3$

STEP 10

Divide both sides of the equation by 4 to solve for $x$ .
$x = \frac{3}{4}$
The solutions to the equations are:
1. For $8^{x}=4$ , $x=\frac{2}{3}$ .
2. For $16^{x}=8$ , $x=\frac{3}{4}$ .

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Question Solve for $x$ in the equations $8^{x}=4$ and $16^{x}=8$ .

Studdy Solution

STEP 1

STEP 2

1. Solve the equation $8^{x}=4$ for $x$ .
2. Solve the equation $16^{x}=8$ for $x$ .

STEP 3

Express both sides of the equation $8^{x}=4$ with the same base.
Since $8 = 2^3$ and $4 = 2^2$ , we can rewrite the equation as:
$(2^3)^x = 2^2$

STEP 4

Apply the power of a power rule to the left side of the equation.
$2^{3x} = 2^2$

STEP 5

Since the bases are now the same, we can equate the exponents.
$3x = 2$

STEP 6

Divide both sides of the equation by 3 to solve for $x$ .
$x = \frac{2}{3}$

STEP 7

Express both sides of the equation $16^{x}=8$ with the same base.
Since $16 = 2^4$ and $8 = 2^3$ , we can rewrite the equation as:
$(2^4)^x = 2^3$

STEP 8

Apply the power of a power rule to the left side of the equation.
$2^{4x} = 2^3$

STEP 9

Since the bases are the same, equate the exponents.
$4x = 3$

STEP 10

Divide both sides of the equation by 4 to solve for $x$ .
$x = \frac{3}{4}$
The solutions to the equations are:
1. For $8^{x}=4$ , $x=\frac{2}{3}$ .
2. For $16^{x}=8$ , $x=\frac{3}{4}$ .

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Question Solve for xxx in the equations 8x=48^{x}=48x=4 and 16x=816^{x}=816x=8.

Studdy Solution

STEP 1

STEP 2

1. Solve the equation 8x=48^{x}=48x=4 for xxx.2. Solve the equation 16x=816^{x}=816x=8 for xxx.

STEP 3

Express both sides of the equation 8x=48^{x}=48x=4 with the same base.Since 8=238 = 2^38=23 and 4=224 = 2^24=22, we can rewrite the equation as:(23)x=22 (2^3)^x = 2^2 (23)x=22

STEP 4

Apply the power of a power rule to the left side of the equation.23x=22 2^{3x} = 2^2 23x=22

STEP 5

Since the bases are now the same, we can equate the exponents.3x=2 3x = 2 3x=2

STEP 6

Divide both sides of the equation by 3 to solve for xxx.x=23 x = \frac{2}{3} x=32​

STEP 7

Express both sides of the equation 16x=816^{x}=816x=8 with the same base.Since 16=2416 = 2^416=24 and 8=238 = 2^38=23, we can rewrite the equation as:(24)x=23 (2^4)^x = 2^3 (24)x=23

STEP 8

Apply the power of a power rule to the left side of the equation.24x=23 2^{4x} = 2^3 24x=23

STEP 9

Since the bases are the same, equate the exponents.4x=3 4x = 3 4x=3

STEP 10

Divide both sides of the equation by 4 to solve for xxx.x=34 x = \frac{3}{4} x=43​The solutions to the equations are:1. For 8x=48^{x}=48x=4, x=23x=\frac{2}{3}x=32​.2. For 16x=816^{x}=816x=8, x=34x=\frac{3}{4}x=43​.

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Question Solve for $x$ in the equations $8^{x}=4$ and $16^{x}=8$ .

1. Solve the equation $8^{x}=4$ for $x$ .
2. Solve the equation $16^{x}=8$ for $x$ .

Express both sides of the equation $8^{x}=4$ with the same base.
Since $8 = 2^3$ and $4 = 2^2$ , we can rewrite the equation as:
$(2^3)^x = 2^2$

Apply the power of a power rule to the left side of the equation.
$2^{3x} = 2^2$

Since the bases are now the same, we can equate the exponents.
$3x = 2$

Divide both sides of the equation by 3 to solve for $x$ .
$x = \frac{2}{3}$

Express both sides of the equation $16^{x}=8$ with the same base.
Since $16 = 2^4$ and $8 = 2^3$ , we can rewrite the equation as:
$(2^4)^x = 2^3$

Apply the power of a power rule to the left side of the equation.
$2^{4x} = 2^3$

Since the bases are the same, equate the exponents.
$4x = 3$

Divide both sides of the equation by 4 to solve for $x$ .
$x = \frac{3}{4}$
The solutions to the equations are:
1. For $8^{x}=4$ , $x=\frac{2}{3}$ .
2. For $16^{x}=8$ , $x=\frac{3}{4}$ .