Solve for x y=(1/3)^x

$y = {(\frac{1}{3})}^{x}$

Step 1

Rewrite the equation as ${(\frac{1}{3})}^{x} = y$ .

${(\frac{1}{3})}^{x} = y$

Step 2

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

$\ln ({(\frac{1}{3})}^{x}) = \ln (y)$

Step 3

Expand the left side.

Expand $\ln ({(\frac{1}{3})}^{x})$ by moving $x$ outside the logarithm.

$x \ln (\frac{1}{3}) = \ln (y)$

Rewrite $\ln (\frac{1}{3})$ as $\ln (1) - \ln (3)$ .

$x (\ln (1) - \ln (3)) = \ln (y)$

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

$x (0 - \ln (3)) = \ln (y)$

Subtract $\ln (3)$ from $0$ .

$x (- \ln (3)) = \ln (y)$

Step 4

Simplify the left side.

Reorder factors in $x (- \ln (3))$ .

$- x \ln (3) = \ln (y)$

Step 5

Divide each term in $- x \ln (3) = \ln (y)$ by $- 1 \ln (3)$ and simplify.

Divide each term in $- x \ln (3) = \ln (y)$ by $- 1 \ln (3)$ .

$\frac{- x \ln (3)}{- 1 \ln (3)} = \frac{\ln (y)}{- 1 \ln (3)}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{x \ln (3)}{\ln (3)} = \frac{\ln (y)}{- 1 \ln (3)}$

Cancel the common factor of $\ln (3)$ .

Cancel the common factor.

$\frac{x \ln (3)}{\ln (3)} = \frac{\ln (y)}{- 1 \ln (3)}$

Divide $x$ by $1$ .

$x = \frac{\ln (y)}{- 1 \ln (3)}$

Simplify the right side.

Move the negative in front of the fraction.

$x = - \frac{\ln (y)}{\ln (3)}$