Evaluate 125^(7x-2)=150

$125^{7 x - 2} = 150$

Step 1

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

$\ln (125^{7 x - 2}) = \ln (150)$

Step 2

Expand $\ln (125^{7 x - 2})$ by moving $7 x - 2$ outside the logarithm.

$(7 x - 2) \ln (125) = \ln (150)$

Step 3

Simplify the left side.

Apply the distributive property.

$7 x \ln (125) - 2 \ln (125) = \ln (150)$

Step 4

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

$7 x \ln (125) - 2 \ln (125) - \ln (150) = 0$

Step 5

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

Add $2 \ln (125)$ to both sides of the equation.

$7 x \ln (125) - \ln (150) = 2 \ln (125)$

Add $\ln (150)$ to both sides of the equation.

$7 x \ln (125) = 2 \ln (125) + \ln (150)$

Step 6

Divide each term in $7 x \ln (125) = 2 \ln (125) + \ln (150)$ by $7 \ln (125)$ and simplify.

Divide each term in $7 x \ln (125) = 2 \ln (125) + \ln (150)$ by $7 \ln (125)$ .

$\frac{7 x \ln (125)}{7 \ln (125)} = \frac{2 \ln (125)}{7 \ln (125)} + \frac{\ln (150)}{7 \ln (125)}$

Simplify the left side.

Cancel the common factor of $7$ .

Cancel the common factor.

$\frac{7 x \ln (125)}{7 \ln (125)} = \frac{2 \ln (125)}{7 \ln (125)} + \frac{\ln (150)}{7 \ln (125)}$

Rewrite the expression.

$\frac{x \ln (125)}{\ln (125)} = \frac{2 \ln (125)}{7 \ln (125)} + \frac{\ln (150)}{7 \ln (125)}$

Cancel the common factor of $\ln (125)$ .

Cancel the common factor.

$\frac{x \ln (125)}{\ln (125)} = \frac{2 \ln (125)}{7 \ln (125)} + \frac{\ln (150)}{7 \ln (125)}$

Divide $x$ by $1$ .

$x = \frac{2 \ln (125)}{7 \ln (125)} + \frac{\ln (150)}{7 \ln (125)}$

Simplify the right side.

Cancel the common factor of $\ln (125)$ .

Cancel the common factor.

$x = \frac{2 \ln (125)}{7 \ln (125)} + \frac{\ln (150)}{7 \ln (125)}$

Rewrite the expression.

$x = \frac{2}{7} + \frac{\ln (150)}{7 \ln (125)}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$x = \frac{2}{7} + \frac{\ln (150)}{7 \ln (125)}$

Decimal Form:

$x = 0.43396584 \dots$

Evaluate 125^(7x-2)=150

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