Solve for t 500=1562.5*10^(-0.1t)

$500 = 1562.5 \cdot 10^{- 0.1 t}$

Step 1

Rewrite the equation as $1562.5 \cdot 10^{- 0.1 t} = 500$ .

$1562.5 \cdot 10^{- 0.1 t} = 500$

Step 2

Divide each term in $(1562.5 \cdot 10^{- 0.1 t}) = 500$ by $1562.5$ and simplify.

Divide each term in $(1562.5 \cdot 10^{- 0.1 t}) = 500$ by $1562.5$ .

$\frac{(1562.5 \cdot 10^{- 0.1 t})}{1562.5} = \frac{500}{1562.5}$

Simplify the left side.

Cancel the common factor of $1562.5$ .

Cancel the common factor.

$\frac{1562.5 \cdot 10^{- 0.1 t}}{1562.5} = \frac{500}{1562.5}$

Divide $10^{- 0.1 t}$ by $1$ .

$10^{- 0.1 t} = \frac{500}{1562.5}$

Simplify the right side.

Divide $500$ by $1562.5$ .

$10^{- 0.1 t} = 0.32$

Step 3

Take the base $10$ logarithm of both sides of the equation to remove the variable from the exponent.

$\log (10^{- 0.1 t}) = \log (0.32)$

Step 4

Expand the left side.

Expand $\log (10^{- 0.1 t})$ by moving $- 0.1 t$ outside the logarithm.

$- 0.1 t \log (10) = \log (0.32)$

Logarithm base $10$ of $10$ is $1$ .

$- 0.1 t \cdot 1 = \log (0.32)$

Multiply $- 0.1$ by $1$ .

$- 0.1 t = \log (0.32)$

Step 5

Simplify the right side.

Log base $10$ of $0.32$ is approximately $- 0.49485002$ .

$- 0.1 t = - 0.49485002$

Step 6

Divide each term in $- 0.1 t = - 0.49485002$ by $- 0.1$ and simplify.

Divide each term in $- 0.1 t = - 0.49485002$ by $- 0.1$ .

$\frac{- 0.1 t}{- 0.1} = \frac{- 0.49485002}{- 0.1}$

Simplify the left side.

Cancel the common factor of $- 0.1$ .

Cancel the common factor.

$\frac{- 0.1 t}{- 0.1} = \frac{- 0.49485002}{- 0.1}$

Divide $t$ by $1$ .

$t = \frac{- 0.49485002}{- 0.1}$

Simplify the right side.

Divide $- 0.49485002$ by $- 0.1$ .

$t = 4.94850021$