Precalculus Examples

2^{x + 3} = 3

$2^{x + 3} = 3$

Step 1

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

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

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

Step 2

Expand

ln (2^{x + 3})

$\ln (2^{x + 3})$ by moving

x + 3

$x + 3$ outside the logarithm.

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

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

Step 3

Simplify the left side.

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Step 3.1

Apply the distributive property.

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

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

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

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

Step 4

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

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

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

Step 5

Move all terms not containing

x

$x$ to the right side of the equation.

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Step 5.1

Subtract

3 ln (2)

$3 \ln (2)$ from both sides of the equation.

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

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

Step 5.2

Add

ln (3)

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

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

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

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

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

Step 6

Divide each term in

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

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

ln (2)

$\ln (2)$ and simplify.

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Step 6.1

Divide each term in

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

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

ln (2)

$\ln (2)$ .

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

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

Step 6.2

Simplify the left side.

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Step 6.2.1

Cancel the common factor of

ln (2)

$\ln (2)$ .

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Step 6.2.1.1

Cancel the common factor.

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

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

Step 6.2.1.2

Divide

x

$x$ by

1

$1$ .

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

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

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

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

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

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

Step 6.3

Simplify the right side.

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Step 6.3.1

Cancel the common factor of

ln (2)

$\ln (2)$ .

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Step 6.3.1.1

Cancel the common factor.

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

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

Step 6.3.1.2

Divide

- 3

$- 3$ by

1

$1$ .

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