Precalculus Examples

$4^{2 x - 1} = 3^{2 x - 3}$

Step 1

Take the log of both sides of the equation.

$\ln (4^{2 x - 1}) = \ln (3^{2 x - 3})$

Step 2

Expand $\ln (4^{2 x - 1})$ by moving $2 x - 1$ outside the logarithm.

$(2 x - 1) \ln (4) = \ln (3^{2 x - 3})$

Step 3

Expand $\ln (3^{2 x - 3})$ by moving $2 x - 3$ outside the logarithm.

$(2 x - 1) \ln (4) = (2 x - 3) \ln (3)$

Step 4

Solve the equation for $x$ .

Tap for more steps...

Step 4.1

Simplify $(2 x - 1) \ln (4)$ .

Tap for more steps...

Step 4.1.1

Rewrite.

$0 + 0 + (2 x - 1) \ln (4) = (2 x - 3) \ln (3)$

Step 4.1.2

Simplify by adding zeros.

$(2 x - 1) \ln (4) = (2 x - 3) \ln (3)$

Step 4.1.3

Apply the distributive property.

$2 x \ln (4) - 1 \ln (4) = (2 x - 3) \ln (3)$

Step 4.1.4

Rewrite $- 1 \ln (4)$ as $- \ln (4)$ .

$2 x \ln (4) - \ln (4) = (2 x - 3) \ln (3)$

Step 4.2

Apply the distributive property.

$2 x \ln (4) - \ln (4) = 2 x \ln (3) - 3 \ln (3)$

Step 4.3

Subtract $2 x \ln (3)$ from both sides of the equation.

$2 x \ln (4) - \ln (4) - 2 x \ln (3) = - 3 \ln (3)$

Step 4.4

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

$2 x \ln (4) - 2 x \ln (3) = - 3 \ln (3) + \ln (4)$

Step 4.5

Factor $2 x$ out of $2 x \ln (4) - 2 x \ln (3)$ .

Tap for more steps...

Step 4.5.1

Factor $2 x$ out of $- 2 x \ln (3)$ .

$2 x \ln (4) + 2 x (- 1 \ln (3)) = - 3 \ln (3) + \ln (4)$

Step 4.5.2

Factor $2 x$ out of $2 x \ln (4) + 2 x (- 1 \ln (3))$ .

$2 x (\ln (4) - 1 \ln (3)) = - 3 \ln (3) + \ln (4)$

Step 4.6

Rewrite $- 1 \ln (3)$ as $- \ln (3)$ .

$2 x (\ln (4) - \ln (3)) = - 3 \ln (3) + \ln (4)$

Step 4.7

Divide each term in $2 x (\ln (4) - \ln (3)) = - 3 \ln (3) + \ln (4)$ by $2 (\ln (4) - \ln (3))$ and simplify.

Tap for more steps...

Step 4.7.1

Divide each term in $2 x (\ln (4) - \ln (3)) = - 3 \ln (3) + \ln (4)$ by $2 (\ln (4) - \ln (3))$ .

$\frac{2 x (\ln (4) - \ln (3))}{2 (\ln (4) - \ln (3))} = \frac{- 3 \ln (3)}{2 (\ln (4) - \ln (3))} + \frac{\ln (4)}{2 (\ln (4) - \ln (3))}$

Step 4.7.2

Simplify the left side.

Tap for more steps...

Step 4.7.2.1

Cancel the common factor of $2$ .

Tap for more steps...

Step 4.7.2.1.1

Cancel the common factor.

$\frac{2 x (\ln (4) - \ln (3))}{2 (\ln (4) - \ln (3))} = \frac{- 3 \ln (3)}{2 (\ln (4) - \ln (3))} + \frac{\ln (4)}{2 (\ln (4) - \ln (3))}$

Step 4.7.2.1.2

Rewrite the expression.

$\frac{x (\ln (4) - \ln (3))}{\ln (4) - \ln (3)} = \frac{- 3 \ln (3)}{2 (\ln (4) - \ln (3))} + \frac{\ln (4)}{2 (\ln (4) - \ln (3))}$

Step 4.7.2.2

Cancel the common factor of $\ln (4) - \ln (3)$ .

Tap for more steps...

Step 4.7.2.2.1

Cancel the common factor.

$\frac{x (\ln (4) - \ln (3))}{\ln (4) - \ln (3)} = \frac{- 3 \ln (3)}{2 (\ln (4) - \ln (3))} + \frac{\ln (4)}{2 (\ln (4) - \ln (3))}$

Step 4.7.2.2.2

Divide $x$ by $1$ .

$x = \frac{- 3 \ln (3)}{2 (\ln (4) - \ln (3))} + \frac{\ln (4)}{2 (\ln (4) - \ln (3))}$

Step 4.7.3

Simplify the right side.

Tap for more steps...

Step 4.7.3.1

Move the negative in front of the fraction.

$x = - \frac{3 \ln (3)}{2 (\ln (4) - \ln (3))} + \frac{\ln (4)}{2 (\ln (4) - \ln (3))}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{3 \ln (3)}{2 (\ln (4) - \ln (3))} + \frac{\ln (4)}{2 (\ln (4) - \ln (3))}$

Decimal Form:

$x = - 3.31884167 \dots$