Solve for x 7^(2x-1)=316

$7^{2 x - 1} = 316$

Step 1

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

$\ln (7^{2 x - 1}) = \ln (316)$

Step 2

Expand $\ln (7^{2 x - 1})$ by moving $2 x - 1$ outside the logarithm.

$(2 x - 1) \ln (7) = \ln (316)$

Step 3

Simplify the left side.

Simplify $(2 x - 1) \ln (7)$ .

Apply the distributive property.

$2 x \ln (7) - 1 \ln (7) = \ln (316)$

Rewrite $- 1 \ln (7)$ as $- \ln (7)$ .

$2 x \ln (7) - \ln (7) = \ln (316)$

Step 4

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

$2 x \ln (7) - \ln (7) - \ln (316) = 0$

Step 5

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

Add $\ln (7)$ to both sides of the equation.

$2 x \ln (7) - \ln (316) = \ln (7)$

Add $\ln (316)$ to both sides of the equation.

$2 x \ln (7) = \ln (7) + \ln (316)$

Step 6

Divide each term in $2 x \ln (7) = \ln (7) + \ln (316)$ by $2 \ln (7)$ and simplify.

Divide each term in $2 x \ln (7) = \ln (7) + \ln (316)$ by $2 \ln (7)$ .

$\frac{2 x \ln (7)}{2 \ln (7)} = \frac{\ln (7)}{2 \ln (7)} + \frac{\ln (316)}{2 \ln (7)}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x \ln (7)}{2 \ln (7)} = \frac{\ln (7)}{2 \ln (7)} + \frac{\ln (316)}{2 \ln (7)}$

Rewrite the expression.

$\frac{x \ln (7)}{\ln (7)} = \frac{\ln (7)}{2 \ln (7)} + \frac{\ln (316)}{2 \ln (7)}$

Cancel the common factor of $\ln (7)$ .

Cancel the common factor.

$\frac{x \ln (7)}{\ln (7)} = \frac{\ln (7)}{2 \ln (7)} + \frac{\ln (316)}{2 \ln (7)}$

Divide $x$ by $1$ .

$x = \frac{\ln (7)}{2 \ln (7)} + \frac{\ln (316)}{2 \ln (7)}$

Simplify the right side.

Cancel the common factor of $\ln (7)$ .

Cancel the common factor.

$x = \frac{\ln (7)}{2 \ln (7)} + \frac{\ln (316)}{2 \ln (7)}$

Rewrite the expression.

$x = \frac{1}{2} + \frac{\ln (316)}{2 \ln (7)}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$x = \frac{1}{2} + \frac{\ln (316)}{2 \ln (7)}$

Decimal Form:

$x = 1.97893319 \dots$

Solve for x 7^(2x-1)=316

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