Evaluate Using the Given Value 4000(1.5^x)=25 , 0

$4000 ({1.5}^{x}) = 25$ , $0$

Step 1

Divide each term in $(4000 \cdot {1.5}^{x}) = 25$ by $4000$ and simplify.

Divide each term in $(4000 \cdot {1.5}^{x}) = 25$ by $4000$ .

$\frac{(4000 \cdot {1.5}^{x})}{4000} = \frac{25}{4000}$

Simplify the left side.

Cancel the common factor of $4000$ .

Cancel the common factor.

$\frac{4000 \cdot {1.5}^{x}}{4000} = \frac{25}{4000}$

Divide ${1.5}^{x}$ by $1$ .

${1.5}^{x} = \frac{25}{4000}$

Simplify the right side.

Cancel the common factor of $25$ and $4000$ .

Factor $25$ out of $25$ .

${1.5}^{x} = \frac{25 (1)}{4000}$

Cancel the common factors.

Factor $25$ out of $4000$ .

${1.5}^{x} = \frac{25 \cdot 1}{25 \cdot 160}$

Cancel the common factor.

${1.5}^{x} = \frac{25 \cdot 1}{25 \cdot 160}$

Rewrite the expression.

${1.5}^{x} = \frac{1}{160}$

Step 2

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

$\ln ({1.5}^{x}) = \ln (\frac{1}{160})$

Step 3

Expand $\ln ({1.5}^{x})$ by moving $x$ outside the logarithm.

$x \ln (1.5) = \ln (\frac{1}{160})$

Step 4

Divide each term in $x \ln (1.5) = \ln (\frac{1}{160})$ by $\ln (1.5)$ and simplify.

Divide each term in $x \ln (1.5) = \ln (\frac{1}{160})$ by $\ln (1.5)$ .

$\frac{x \ln (1.5)}{\ln (1.5)} = \frac{\ln (\frac{1}{160})}{\ln (1.5)}$

Simplify the left side.

Cancel the common factor of $\ln (1.5)$ .

Cancel the common factor.

$\frac{x \ln (1.5)}{\ln (1.5)} = \frac{\ln (\frac{1}{160})}{\ln (1.5)}$

Divide $x$ by $1$ .

$x = \frac{\ln (\frac{1}{160})}{\ln (1.5)}$

Step 5

Replace the variable $x$ with $\frac{\ln (\frac{1}{160})}{\ln (1.5)}$ in the expression.

$0$