Evaluate Using the Given Value 2 , 401=7^(6-2x)

$2$ , $401 = 7^{6 - 2 x}$

Step 1

Rewrite the equation as $7^{6 - 2 x} = 401$ .

$7^{6 - 2 x} = 401$

Step 2

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

$\ln (7^{6 - 2 x}) = \ln (401)$

Step 3

Expand $\ln (7^{6 - 2 x})$ by moving $6 - 2 x$ outside the logarithm.

$(6 - 2 x) \ln (7) = \ln (401)$

Step 4

Simplify the left side.

Apply the distributive property.

$6 \ln (7) - 2 x \ln (7) = \ln (401)$

Step 5

Reorder $6 \ln (7)$ and $- 2 x \ln (7)$ .

$- 2 x \ln (7) + 6 \ln (7) = \ln (401)$

Step 6

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

$- 2 x \ln (7) + 6 \ln (7) - \ln (401) = 0$

Step 7

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

Subtract $6 \ln (7)$ from both sides of the equation.

$- 2 x \ln (7) - \ln (401) = - 6 \ln (7)$

Add $\ln (401)$ to both sides of the equation.

$- 2 x \ln (7) = - 6 \ln (7) + \ln (401)$

Step 8

Divide each term in $- 2 x \ln (7) = - 6 \ln (7) + \ln (401)$ by $- 2 \ln (7)$ and simplify.

Divide each term in $- 2 x \ln (7) = - 6 \ln (7) + \ln (401)$ by $- 2 \ln (7)$ .

$\frac{- 2 x \ln (7)}{- 2 \ln (7)} = \frac{- 6 \ln (7)}{- 2 \ln (7)} + \frac{\ln (401)}{- 2 \ln (7)}$

Simplify the left side.

Cancel the common factor of $- 2$ .

Cancel the common factor.

$\frac{- 2 x \ln (7)}{- 2 \ln (7)} = \frac{- 6 \ln (7)}{- 2 \ln (7)} + \frac{\ln (401)}{- 2 \ln (7)}$

Rewrite the expression.

$\frac{x \ln (7)}{\ln (7)} = \frac{- 6 \ln (7)}{- 2 \ln (7)} + \frac{\ln (401)}{- 2 \ln (7)}$

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

Cancel the common factor.

$\frac{x \ln (7)}{\ln (7)} = \frac{- 6 \ln (7)}{- 2 \ln (7)} + \frac{\ln (401)}{- 2 \ln (7)}$

Divide $x$ by $1$ .

$x = \frac{- 6 \ln (7)}{- 2 \ln (7)} + \frac{\ln (401)}{- 2 \ln (7)}$

Simplify the right side.

Simplify each term.

Cancel the common factor of $- 6$ and $- 2$ .

Factor $- 2$ out of $- 6 \ln (7)$ .

$x = \frac{- 2 (3 \ln (7))}{- 2 \ln (7)} + \frac{\ln (401)}{- 2 \ln (7)}$

Cancel the common factors.

Factor $- 2$ out of $- 2 \ln (7)$ .

$x = \frac{- 2 (3 \ln (7))}{- 2 \ln (7)} + \frac{\ln (401)}{- 2 \ln (7)}$

Cancel the common factor.

$x = \frac{- 2 (3 \ln (7))}{- 2 \ln (7)} + \frac{\ln (401)}{- 2 \ln (7)}$

Rewrite the expression.

$x = \frac{3 \ln (7)}{\ln (7)} + \frac{\ln (401)}{- 2 \ln (7)}$

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

Cancel the common factor.

$x = \frac{3 \ln (7)}{\ln (7)} + \frac{\ln (401)}{- 2 \ln (7)}$

Divide $3$ by $1$ .

$x = 3 + \frac{\ln (401)}{- 2 \ln (7)}$

Move the negative in front of the fraction.

$x = 3 - \frac{\ln (401)}{2 \ln (7)}$

Step 9

Replace the variable $x$ with $3 - \frac{\ln (401)}{2 \ln (7)}$ in the expression.

$2$

Evaluate Using the Given Value 2 , 401=7^(6-2x)

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