Solve for x 12^(2x-8)=15

$12^{2 x - 8} = 15$

Step 1

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

$\ln (12^{2 x - 8}) = \ln (15)$

Step 2

Expand $\ln (12^{2 x - 8})$ by moving $2 x - 8$ outside the logarithm.

$(2 x - 8) \ln (12) = \ln (15)$

Step 3

Simplify the left side.

Apply the distributive property.

$2 x \ln (12) - 8 \ln (12) = \ln (15)$

Step 4

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

$2 x \ln (12) - 8 \ln (12) - \ln (15) = 0$

Step 5

Move all terms not containing $x$ to the right side of the equation.

Add $8 \ln (12)$ to both sides of the equation.

$2 x \ln (12) - \ln (15) = 8 \ln (12)$

Add $\ln (15)$ to both sides of the equation.

$2 x \ln (12) = 8 \ln (12) + \ln (15)$

Step 6

Divide each term in $2 x \ln (12) = 8 \ln (12) + \ln (15)$ by $2 \ln (12)$ and simplify.

Divide each term in $2 x \ln (12) = 8 \ln (12) + \ln (15)$ by $2 \ln (12)$ .

$\frac{2 x \ln (12)}{2 \ln (12)} = \frac{8 \ln (12)}{2 \ln (12)} + \frac{\ln (15)}{2 \ln (12)}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x \ln (12)}{2 \ln (12)} = \frac{8 \ln (12)}{2 \ln (12)} + \frac{\ln (15)}{2 \ln (12)}$

Rewrite the expression.

$\frac{x \ln (12)}{\ln (12)} = \frac{8 \ln (12)}{2 \ln (12)} + \frac{\ln (15)}{2 \ln (12)}$

Cancel the common factor of $\ln (12)$ .

Cancel the common factor.

$\frac{x \ln (12)}{\ln (12)} = \frac{8 \ln (12)}{2 \ln (12)} + \frac{\ln (15)}{2 \ln (12)}$

Divide $x$ by $1$ .

$x = \frac{8 \ln (12)}{2 \ln (12)} + \frac{\ln (15)}{2 \ln (12)}$

Simplify the right side.

Simplify each term.

Cancel the common factor of $8$ and $2$ .

Factor $2$ out of $8 \ln (12)$ .

$x = \frac{2 (4 \ln (12))}{2 \ln (12)} + \frac{\ln (15)}{2 \ln (12)}$

Cancel the common factors.

Factor $2$ out of $2 \ln (12)$ .

$x = \frac{2 (4 \ln (12))}{2 \ln (12)} + \frac{\ln (15)}{2 \ln (12)}$

Cancel the common factor.

$x = \frac{2 (4 \ln (12))}{2 \ln (12)} + \frac{\ln (15)}{2 \ln (12)}$

Rewrite the expression.

$x = \frac{4 \ln (12)}{\ln (12)} + \frac{\ln (15)}{2 \ln (12)}$

Cancel the common factor of $\ln (12)$ .

Cancel the common factor.

$x = \frac{4 \ln (12)}{\ln (12)} + \frac{\ln (15)}{2 \ln (12)}$

Divide $4$ by $1$ .

$x = 4 + \frac{\ln (15)}{2 \ln (12)}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$x = 4 + \frac{\ln (15)}{2 \ln (12)}$

Decimal Form:

$x = 4.54489978 \dots$

Solve for x 12^(2x-8)=15

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