Solve for a (4x^3)^a=16x^6

${(4 x^{3})}^{a} = 16 x^{6}$

Step 1

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

$\ln ({(4 x^{3})}^{a}) = \ln (16 x^{6})$

Step 2

Expand the left side.

Expand $\ln ({(4 x^{3})}^{a})$ by moving $a$ outside the logarithm.

$a \ln (4 x^{3}) = \ln (16 x^{6})$

Rewrite $\ln (4 x^{3})$ as $\ln (4) + \ln (x^{3})$ .

$a (\ln (4) + \ln (x^{3})) = \ln (16 x^{6})$

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

$a (\ln (4) + 3 \ln (x)) = \ln (16 x^{6})$

Step 3

Simplify the left side.

Simplify $a (\ln (4) + 3 \ln (x))$ .

Apply the distributive property.

$a \ln (4) + a (3 \ln (x)) = \ln (16 x^{6})$

Rewrite using the commutative property of multiplication.

$a \ln (4) + 3 a \ln (x) = \ln (16 x^{6})$

Step 4

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

$a \ln (4) + 3 a \ln (x) - \ln (16 x^{6}) = 0$

Step 5

Add $\ln (16 x^{6})$ to both sides of the equation.

$a \ln (4) + 3 a \ln (x) = \ln (16 x^{6})$

Step 6

Factor $a$ out of $a \ln (4) + 3 a \ln (x)$ .

Factor $a$ out of $a \ln (4)$ .

$a (\ln (4)) + 3 a \ln (x) = \ln (16 x^{6})$

Factor $a$ out of $3 a \ln (x)$ .

$a (\ln (4)) + a (3 \ln (x)) = \ln (16 x^{6})$

Factor $a$ out of $a (\ln (4)) + a (3 \ln (x))$ .

$a (\ln (4) + 3 \ln (x)) = \ln (16 x^{6})$

Step 7

Divide each term in $a (\ln (4) + 3 \ln (x)) = \ln (16 x^{6})$ by $\ln (4) + 3 \ln (x)$ and simplify.

Divide each term in $a (\ln (4) + 3 \ln (x)) = \ln (16 x^{6})$ by $\ln (4) + 3 \ln (x)$ .

$\frac{a (\ln (4) + 3 \ln (x))}{\ln (4) + 3 \ln (x)} = \frac{\ln (16 x^{6})}{\ln (4) + 3 \ln (x)}$

Simplify the left side.

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

Cancel the common factor.

$\frac{a (\ln (4) + 3 \ln (x))}{\ln (4) + 3 \ln (x)} = \frac{\ln (16 x^{6})}{\ln (4) + 3 \ln (x)}$

Divide $a$ by $1$ .

$a = \frac{\ln (16 x^{6})}{\ln (4) + 3 \ln (x)}$