Solve for x 2400=7500(10)^(-x)

$2400 = 7500 {(10)}^{- x}$

Step 1

Rewrite the equation as $7500 \cdot 10^{- x} = 2400$ .

$7500 \cdot 10^{- x} = 2400$

Step 2

Divide each term in $(7500 \cdot 10^{- x}) = 2400$ by $7500$ and simplify.

Divide each term in $(7500 \cdot 10^{- x}) = 2400$ by $7500$ .

$\frac{(7500 \cdot 10^{- x})}{7500} = \frac{2400}{7500}$

Simplify the left side.

Cancel the common factor of $7500$ .

Cancel the common factor.

$\frac{7500 \cdot 10^{- x}}{7500} = \frac{2400}{7500}$

Divide $10^{- x}$ by $1$ .

$10^{- x} = \frac{2400}{7500}$

Simplify the right side.

Cancel the common factor of $2400$ and $7500$ .

Factor $300$ out of $2400$ .

$10^{- x} = \frac{300 (8)}{7500}$

Cancel the common factors.

Factor $300$ out of $7500$ .

$10^{- x} = \frac{300 \cdot 8}{300 \cdot 25}$

Cancel the common factor.

$10^{- x} = \frac{300 \cdot 8}{300 \cdot 25}$

Rewrite the expression.

$10^{- x} = \frac{8}{25}$

Step 3

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

$\log (10^{- x}) = \log (\frac{8}{25})$

Step 4

Expand the left side.

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

$- x \log (10) = \log (\frac{8}{25})$

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

$- x \cdot 1 = \log (\frac{8}{25})$

Multiply $- 1$ by $1$ .

$- x = \log (\frac{8}{25})$

Step 5

Divide each term in $- x = \log (\frac{8}{25})$ by $- 1$ and simplify.

Divide each term in $- x = \log (\frac{8}{25})$ by $- 1$ .

$\frac{- x}{- 1} = \frac{\log (\frac{8}{25})}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{\log (\frac{8}{25})}{- 1}$

Divide $x$ by $1$ .

$x = \frac{\log (\frac{8}{25})}{- 1}$

Simplify the right side.

Move the negative one from the denominator of $\frac{\log (\frac{8}{25})}{- 1}$ .

$x = - 1 \cdot \log (\frac{8}{25})$

Rewrite $- 1 \cdot \log (\frac{8}{25})$ as $- \log (\frac{8}{25})$ .

$x = - \log (\frac{8}{25})$

Step 6

The result can be shown in multiple forms.

Exact Form:

$x = - \log (\frac{8}{25})$

Decimal Form:

$x = 0.49485002 \dots$

Solve for x 2400=7500(10)^(-x)

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