Precalculus Examples

$36^{x} - 4 \cdot 6^{x} = - 4$

Step 1

Add $4$ to both sides of the equation.

$36^{x} - 4 \cdot 6^{x} + 4 = 0$

Step 2

Factor the left side of the equation.

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Step 2.1

Rewrite $36^{x}$ as ${(6^{2})}^{x}$ .

${(6^{2})}^{x} - 4 \cdot 6^{x} + 4 = 0$

Step 2.2

Rewrite ${(6^{2})}^{x}$ as ${(6^{x})}^{2}$ .

${(6^{x})}^{2} - 4 \cdot 6^{x} + 4 = 0$

Step 2.3

Let $u = 6^{x}$ . Substitute $u$ for all occurrences of $6^{x}$ .

$u^{2} - 4 u + 4 = 0$

Step 2.4

Factor using the perfect square rule.

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Step 2.4.1

Rewrite $4$ as $2^{2}$ .

$u^{2} - 4 u + 2^{2} = 0$

Step 2.4.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$4 u = 2 \cdot u \cdot 2$

Step 2.4.3

Rewrite the polynomial.

$u^{2} - 2 \cdot u \cdot 2 + 2^{2} = 0$

Step 2.4.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = u$ and $b = 2$ .

${(u - 2)}^{2} = 0$

Step 2.5

Replace all occurrences of $u$ with $6^{x}$ .

${(6^{x} - 2)}^{2} = 0$

Step 3

Set the $6^{x} - 2$ equal to $0$ .

$6^{x} - 2 = 0$

Step 4

Solve for $x$ .

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Step 4.1

Add $2$ to both sides of the equation.

$6^{x} = 2$

Step 4.2

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

$\ln (6^{x}) = \ln (2)$

Step 4.3

Expand $\ln (6^{x})$ by moving $x$ outside the logarithm.

$x \ln (6) = \ln (2)$

Step 4.4

Divide each term in $x \ln (6) = \ln (2)$ by $\ln (6)$ and simplify.

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Step 4.4.1

Divide each term in $x \ln (6) = \ln (2)$ by $\ln (6)$ .

$\frac{x \ln (6)}{\ln (6)} = \frac{\ln (2)}{\ln (6)}$

Step 4.4.2

Simplify the left side.

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Step 4.4.2.1

Cancel the common factor of $\ln (6)$ .

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Step 4.4.2.1.1

Cancel the common factor.

$\frac{x \ln (6)}{\ln (6)} = \frac{\ln (2)}{\ln (6)}$

Step 4.4.2.1.2

Divide $x$ by $1$ .

$x = \frac{\ln (2)}{\ln (6)}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = \frac{\ln (2)}{\ln (6)}$

Decimal Form:

$x = 0.38685280 \dots$