Solve for x 7^(2-x)+7^(-x)=50

$7^{2 - x} + 7^{- x} = 50$

Step 1

Factor out $7^{- 1 x}$ from the expression.

$7^{- 1 x} (49 + 1)$

Step 2

Rewrite $- 1 x$ as $- x$ .

$7^{- x} (49 + 1) = 50$

Step 3

Add $49$ and $1$ .

$7^{- x} \cdot 50 = 50$

Step 4

Move $50$ to the left of $7^{- x}$ .

$50 \cdot 7^{- x} = 50$

Step 5

Divide each term in $(50 \cdot 7^{- x}) = 50$ by $50$ and simplify.

Divide each term in $(50 \cdot 7^{- x}) = 50$ by $50$ .

$\frac{(50 \cdot 7^{- x})}{50} = \frac{50}{50}$

Simplify the left side.

Cancel the common factor of $50$ .

Cancel the common factor.

$\frac{50 \cdot 7^{- x}}{50} = \frac{50}{50}$

Divide $7^{- x}$ by $1$ .

$7^{- x} = \frac{50}{50}$

Simplify the right side.

Divide $50$ by $50$ .

$7^{- x} = 1$

Step 6

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

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

Step 7

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

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

Step 8

Simplify the right side.

The natural logarithm of $1$ is $0$ .

$- x \ln (7) = 0$

Step 9

Divide each term in $- x \ln (7) = 0$ by $- \ln (7)$ and simplify.

Divide each term in $- x \ln (7) = 0$ by $- \ln (7)$ .

$\frac{- x \ln (7)}{- \ln (7)} = \frac{0}{- \ln (7)}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{x \ln (7)}{\ln (7)} = \frac{0}{- \ln (7)}$

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

Cancel the common factor.

$\frac{x \ln (7)}{\ln (7)} = \frac{0}{- \ln (7)}$

Divide $x$ by $1$ .

$x = \frac{0}{- \ln (7)}$

Simplify the right side.

Divide $0$ by $- \ln (7)$ .

$x = 0$

Solve for x 7^(2-x)+7^(-x)=50

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