Algebra Examples

$(2^{x} - 3) (2^{x} - 4) = 0$

Step 1

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$2^{x} - 3 = 0$

$2^{x} - 4 = 0$

Step 2

Set $2^{x} - 3$ equal to $0$ and solve for $x$ .

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Step 2.1

Set $2^{x} - 3$ equal to $0$ .

$2^{x} - 3 = 0$

Step 2.2

Solve $2^{x} - 3 = 0$ for $x$ .

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Step 2.2.1

Add $3$ to both sides of the equation.

$2^{x} = 3$

Step 2.2.2

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

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

Step 2.2.3

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

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

Step 2.2.4

Divide each term in $x \ln (2) = \ln (3)$ by $\ln (2)$ and simplify.

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Step 2.2.4.1

Divide each term in $x \ln (2) = \ln (3)$ by $\ln (2)$ .

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

Step 2.2.4.2

Simplify the left side.

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Step 2.2.4.2.1

Cancel the common factor of $\ln (2)$ .

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Step 2.2.4.2.1.1

Cancel the common factor.

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

Step 2.2.4.2.1.2

Divide $x$ by $1$ .

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

Step 3

Set $2^{x} - 4$ equal to $0$ and solve for $x$ .

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Step 3.1

Set $2^{x} - 4$ equal to $0$ .

$2^{x} - 4 = 0$

Step 3.2

Solve $2^{x} - 4 = 0$ for $x$ .

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Step 3.2.1

Add $4$ to both sides of the equation.

$2^{x} = 4$

Step 3.2.2

Create equivalent expressions in the equation that all have equal bases.

$2^{x} = 2^{2}$

Step 3.2.3

Since the bases are the same, then two expressions are only equal if the exponents are also equal.

$x = 2$

Step 4

The final solution is all the values that make $(2^{x} - 3) (2^{x} - 4) = 0$ true.

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

Step 5

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$x = 1.58496250 \dots, 2$