Solve for x (2^x-3)(2^x-4)=0

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

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

$2^{x} - 3 = 0$

$2^{x} - 4 = 0$

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

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

$2^{x} - 3 = 0$

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

Add $3$ to both sides of the equation.

$2^{x} = 3$

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

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

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

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

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

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

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

Simplify the left side.

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

Cancel the common factor.

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

Divide $x$ by $1$ .

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

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

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

$2^{x} - 4 = 0$

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

Add $4$ to both sides of the equation.

$2^{x} = 4$

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

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

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

$x = 2$

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

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

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$x = 1.58496250 \dots, 2$