Algebra Examples

$\frac{1}{4} (4^{x - 1}) + 3 = 12$

Step 1

Simplify each term.

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Step 1.1

Combine $\frac{1}{4}$ and $4^{x - 1}$ .

$\frac{4^{x - 1}}{4} + 3 = 12$

Step 1.2

Cancel the common factor of $4^{x - 1}$ and $4$ .

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Step 1.2.1

Factor $4$ out of $4^{x - 1}$ .

$\frac{4 \cdot 4^{x - 2}}{4} + 3 = 12$

Step 1.2.2

Cancel the common factors.

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Step 1.2.2.1

Factor $4$ out of $4$ .

$\frac{4 \cdot 4^{x - 2}}{4 (1)} + 3 = 12$

Step 1.2.2.2

Cancel the common factor.

$\frac{4 \cdot 4^{x - 2}}{4 \cdot 1} + 3 = 12$

Step 1.2.2.3

Rewrite the expression.

$\frac{4^{x - 2}}{1} + 3 = 12$

Step 1.2.2.4

Divide $4^{x - 2}$ by $1$ .

$4^{x - 2} + 3 = 12$

Step 2

Move all terms not containing $x$ to the right side of the equation.

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Step 2.1

Subtract $3$ from both sides of the equation.

$4^{x - 2} = 12 - 3$

Step 2.2

Subtract $3$ from $12$ .

$4^{x - 2} = 9$

Step 3

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

$\ln (4^{x - 2}) = \ln (9)$

Step 4

Expand $\ln (4^{x - 2})$ by moving $x - 2$ outside the logarithm.

$(x - 2) \ln (4) = \ln (9)$

Step 5

Simplify the left side.

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Step 5.1

Apply the distributive property.

$x \ln (4) - 2 \ln (4) = \ln (9)$

Step 6

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

$x \ln (4) - 2 \ln (4) - \ln (9) = 0$

Step 7

Move all terms not containing $x$ to the right side of the equation.

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Step 7.1

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

$x \ln (4) - \ln (9) = 2 \ln (4)$

Step 7.2

Add $\ln (9)$ to both sides of the equation.

$x \ln (4) = 2 \ln (4) + \ln (9)$

Step 8

Divide each term in $x \ln (4) = 2 \ln (4) + \ln (9)$ by $\ln (4)$ and simplify.

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Step 8.1

Divide each term in $x \ln (4) = 2 \ln (4) + \ln (9)$ by $\ln (4)$ .

$\frac{x \ln (4)}{\ln (4)} = \frac{2 \ln (4)}{\ln (4)} + \frac{\ln (9)}{\ln (4)}$

Step 8.2

Simplify the left side.

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Step 8.2.1

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

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Step 8.2.1.1

Cancel the common factor.

$\frac{x \ln (4)}{\ln (4)} = \frac{2 \ln (4)}{\ln (4)} + \frac{\ln (9)}{\ln (4)}$

Step 8.2.1.2

Divide $x$ by $1$ .

$x = \frac{2 \ln (4)}{\ln (4)} + \frac{\ln (9)}{\ln (4)}$

Step 8.3

Simplify the right side.

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Step 8.3.1

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

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Step 8.3.1.1

Cancel the common factor.

$x = \frac{2 \ln (4)}{\ln (4)} + \frac{\ln (9)}{\ln (4)}$

Step 8.3.1.2

Divide $2$ by $1$ .

$x = 2 + \frac{\ln (9)}{\ln (4)}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$x = 2 + \frac{\ln (9)}{\ln (4)}$

Decimal Form:

$x = 3.58496250 \dots$