Examples

2^{2 x + 4} = 3

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

Step 1

Take the natural logarithm of both sides of the equation to remove the variable from the exponent.

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

$\ln (2^{2 x + 4}) = \ln (3)$

Step 2

Expand

ln (2^{2 x + 4})

$\ln (2^{2 x + 4})$ by moving

2 x + 4

$2 x + 4$ outside the logarithm.

(2 x + 4) ln (2) = ln (3)

$(2 x + 4) \ln (2) = \ln (3)$

Step 3

Simplify the left side.

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Step 3.1

Apply the distributive property.

2 x ln (2) + 4 ln (2) = ln (3)

$2 x \ln (2) + 4 \ln (2) = \ln (3)$

2 x ln (2) + 4 ln (2) = ln (3)

$2 x \ln (2) + 4 \ln (2) = \ln (3)$

Step 4

Move all the terms containing a logarithm to the left side of the equation.

2 x ln (2) + 4 ln (2) - ln (3) = 0

$2 x \ln (2) + 4 \ln (2) - \ln (3) = 0$

Step 5

Move all terms not containing

x

$x$ to the right side of the equation.

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Step 5.1

Subtract

4 ln (2)

$4 \ln (2)$ from both sides of the equation.

2 x ln (2) - ln (3) = - 4 ln (2)

$2 x \ln (2) - \ln (3) = - 4 \ln (2)$

Step 5.2

Add

ln (3)

$\ln (3)$ to both sides of the equation.

2 x ln (2) = - 4 ln (2) + ln (3)

$2 x \ln (2) = - 4 \ln (2) + \ln (3)$

2 x ln (2) = - 4 ln (2) + ln (3)

$2 x \ln (2) = - 4 \ln (2) + \ln (3)$

Step 6

Divide each term in

2 x ln (2) = - 4 ln (2) + ln (3)

$2 x \ln (2) = - 4 \ln (2) + \ln (3)$ by

2 ln (2)

$2 \ln (2)$ and simplify.

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Step 6.1

Divide each term in

2 x ln (2) = - 4 ln (2) + ln (3)

$2 x \ln (2) = - 4 \ln (2) + \ln (3)$ by

2 ln (2)

$2 \ln (2)$ .

\frac{2 x ln (2)}{2 ln (2)} = \frac{- 4 ln (2)}{2 ln (2)} + \frac{ln (3)}{2 ln (2)}

$\frac{2 x \ln (2)}{2 \ln (2)} = \frac{- 4 \ln (2)}{2 \ln (2)} + \frac{\ln (3)}{2 \ln (2)}$

Step 6.2

Simplify the left side.

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Step 6.2.1

Cancel the common factor of

2

$2$ .

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Step 6.2.1.1

Cancel the common factor.

\frac{2 x ln (2)}{2 ln (2)} = \frac{- 4 ln (2)}{2 ln (2)} + \frac{ln (3)}{2 ln (2)}

$\frac{2 x \ln (2)}{2 \ln (2)} = \frac{- 4 \ln (2)}{2 \ln (2)} + \frac{\ln (3)}{2 \ln (2)}$

Step 6.2.1.2

Rewrite the expression.

\frac{x ln (2)}{ln (2)} = \frac{- 4 ln (2)}{2 ln (2)} + \frac{ln (3)}{2 ln (2)}

$\frac{x \ln (2)}{\ln (2)} = \frac{- 4 \ln (2)}{2 \ln (2)} + \frac{\ln (3)}{2 \ln (2)}$

\frac{x ln (2)}{ln (2)} = \frac{- 4 ln (2)}{2 ln (2)} + \frac{ln (3)}{2 ln (2)}

$\frac{x \ln (2)}{\ln (2)} = \frac{- 4 \ln (2)}{2 \ln (2)} + \frac{\ln (3)}{2 \ln (2)}$

Step 6.2.2

Cancel the common factor of

ln (2)

$\ln (2)$ .

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Step 6.2.2.1

Cancel the common factor.

\frac{x ln (2)}{ln (2)} = \frac{- 4 ln (2)}{2 ln (2)} + \frac{ln (3)}{2 ln (2)}

$\frac{x \ln (2)}{\ln (2)} = \frac{- 4 \ln (2)}{2 \ln (2)} + \frac{\ln (3)}{2 \ln (2)}$

Step 6.2.2.2

Divide

x

$x$ by

1

$1$ .

x = \frac{- 4 ln (2)}{2 ln (2)} + \frac{ln (3)}{2 ln (2)}

$x = \frac{- 4 \ln (2)}{2 \ln (2)} + \frac{\ln (3)}{2 \ln (2)}$

x = \frac{- 4 ln (2)}{2 ln (2)} + \frac{ln (3)}{2 ln (2)}

$x = \frac{- 4 \ln (2)}{2 \ln (2)} + \frac{\ln (3)}{2 \ln (2)}$

x = \frac{- 4 ln (2)}{2 ln (2)} + \frac{ln (3)}{2 ln (2)}

$x = \frac{- 4 \ln (2)}{2 \ln (2)} + \frac{\ln (3)}{2 \ln (2)}$

Step 6.3

Simplify the right side.

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Step 6.3.1

Simplify each term.

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Step 6.3.1.1

Cancel the common factor of

- 4

$- 4$ and

2

$2$ .

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Step 6.3.1.1.1

Factor

2

$2$ out of

- 4 ln (2)

$- 4 \ln (2)$ .

x = \frac{2 (- 2 ln (2))}{2 ln (2)} + \frac{ln (3)}{2 ln (2)}

$x = \frac{2 (- 2 \ln (2))}{2 \ln (2)} + \frac{\ln (3)}{2 \ln (2)}$

Step 6.3.1.1.2

Cancel the common factors.

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Step 6.3.1.1.2.1

Factor

2

$2$ out of

2 ln (2)

$2 \ln (2)$ .

x = \frac{2 (- 2 ln (2))}{2 ln (2)} + \frac{ln (3)}{2 ln (2)}

$x = \frac{2 (- 2 \ln (2))}{2 \ln (2)} + \frac{\ln (3)}{2 \ln (2)}$

Step 6.3.1.1.2.2

Cancel the common factor.

x = \frac{2 (- 2 ln (2))}{2 ln (2)} + \frac{ln (3)}{2 ln (2)}

$x = \frac{2 (- 2 \ln (2))}{2 \ln (2)} + \frac{\ln (3)}{2 \ln (2)}$

Step 6.3.1.1.2.3

Rewrite the expression.

x = \frac{- 2 ln (2)}{ln (2)} + \frac{ln (3)}{2 ln (2)}

$x = \frac{- 2 \ln (2)}{\ln (2)} + \frac{\ln (3)}{2 \ln (2)}$

x = \frac{- 2 ln (2)}{ln (2)} + \frac{ln (3)}{2 ln (2)}

$x = \frac{- 2 \ln (2)}{\ln (2)} + \frac{\ln (3)}{2 \ln (2)}$

x = \frac{- 2 ln (2)}{ln (2)} + \frac{ln (3)}{2 ln (2)}

$x = \frac{- 2 \ln (2)}{\ln (2)} + \frac{\ln (3)}{2 \ln (2)}$

Step 6.3.1.2

Cancel the common factor of

ln (2)

$\ln (2)$ .

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Step 6.3.1.2.1

Cancel the common factor.

x = \frac{- 2 ln (2)}{ln (2)} + \frac{ln (3)}{2 ln (2)}

$x = \frac{- 2 \ln (2)}{\ln (2)} + \frac{\ln (3)}{2 \ln (2)}$

Step 6.3.1.2.2

Divide

- 2

$- 2$ by

1

$1$ .

x = - 2 + \frac{ln (3)}{2 ln (2)}