Precalculus Examples

$4^{x + 1} = e^{x - 1}$

Step 1

Take the log of both sides of the equation.

$\ln (4^{x + 1}) = \ln (e^{x - 1})$

Step 2

Expand $\ln (4^{x + 1})$ by moving $x + 1$ outside the logarithm.

$(x + 1) \ln (4) = \ln (e^{x - 1})$

Step 3

Expand $\ln (e^{x - 1})$ by moving $x - 1$ outside the logarithm.

$(x + 1) \ln (4) = (x - 1) \ln (e)$

Step 4

The natural logarithm of $e$ is $1$ .

$(x + 1) \ln (4) = (x - 1) \cdot 1$

Step 5

Multiply $x - 1$ by $1$ .

$(x + 1) \ln (4) = x - 1$

Step 6

Solve the equation for $x$ .

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Step 6.1

Simplify $(x + 1) \ln (4)$ .

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Step 6.1.1

Rewrite.

$0 + 0 + (x + 1) \ln (4) = x - 1$

Step 6.1.2

Simplify by adding zeros.

$(x + 1) \ln (4) = x - 1$

Step 6.1.3

Apply the distributive property.

$x \ln (4) + 1 \ln (4) = x - 1$

Step 6.1.4

Multiply $\ln (4)$ by $1$ .

$x \ln (4) + \ln (4) = x - 1$

Step 6.2

Subtract $x$ from both sides of the equation.

$x \ln (4) + \ln (4) - x = - 1$

Step 6.3

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

$x \ln (4) + \ln (4) = x - 1$

Step 6.4

Subtract $x$ from both sides of the equation.

$x \ln (4) + \ln (4) - x = - 1$

Step 6.5

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

$x \ln (4) - x = - 1 - \ln (4)$

Step 6.6

Factor $x$ out of $x \ln (4) - x$ .

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Step 6.6.1

Factor $x$ out of $x \ln (4)$ .

$x (\ln (4)) - x = - 1 - \ln (4)$

Step 6.6.2

Factor $x$ out of $- x$ .

$x (\ln (4)) + x \cdot - 1 = - 1 - \ln (4)$

Step 6.6.3

Factor $x$ out of $x (\ln (4)) + x \cdot - 1$ .

$x (\ln (4) - 1) = - 1 - \ln (4)$

Step 6.7

Divide each term in $x (\ln (4) - 1) = - 1 - \ln (4)$ by $\ln (4) - 1$ and simplify.

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Step 6.7.1

Divide each term in $x (\ln (4) - 1) = - 1 - \ln (4)$ by $\ln (4) - 1$ .

$\frac{x (\ln (4) - 1)}{\ln (4) - 1} = \frac{- 1}{\ln (4) - 1} + \frac{- \ln (4)}{\ln (4) - 1}$

Step 6.7.2

Simplify the left side.

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Step 6.7.2.1

Cancel the common factor of $\ln (4) - 1$ .

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Step 6.7.2.1.1

Cancel the common factor.

$\frac{x (\ln (4) - 1)}{\ln (4) - 1} = \frac{- 1}{\ln (4) - 1} + \frac{- \ln (4)}{\ln (4) - 1}$

Step 6.7.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 1}{\ln (4) - 1} + \frac{- \ln (4)}{\ln (4) - 1}$

Step 6.7.3

Simplify the right side.

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Step 6.7.3.1

Combine the numerators over the common denominator.

$x = \frac{- 1 - \ln (4)}{\ln (4) - 1}$

Step 6.7.3.2

Rewrite $- 1$ as $- 1 (1)$ .

$x = \frac{- 1 (1) - \ln (4)}{\ln (4) - 1}$

Step 6.7.3.3

Factor $- 1$ out of $- \ln (4)$ .

$x = \frac{- 1 (1) - \ln (4)}{\ln (4) - 1}$

Step 6.7.3.4

Factor $- 1$ out of $- 1 (1) - \ln (4)$ .

$x = \frac{- 1 (1 + \ln (4))}{\ln (4) - 1}$

Step 6.7.3.5

Move the negative in front of the fraction.

$x = - \frac{1 + \ln (4)}{\ln (4) - 1}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{1 + \ln (4)}{\ln (4) - 1}$

Decimal Form:

$x = - 6.17739889 \dots$