Evaluate 500*5^(y/3)=1

$500 \cdot 5^{\frac{y}{3}} = 1$

Step 1

Divide each term in $(500 \cdot 5^{\frac{y}{3}}) = 1$ by $500$ and simplify.

Divide each term in $(500 \cdot 5^{\frac{y}{3}}) = 1$ by $500$ .

$\frac{(500 \cdot 5^{\frac{y}{3}})}{500} = \frac{1}{500}$

Simplify the left side.

Cancel the common factor of $500$ .

Cancel the common factor.

$\frac{500 \cdot 5^{\frac{y}{3}}}{500} = \frac{1}{500}$

Divide $5^{\frac{y}{3}}$ by $1$ .

$5^{\frac{y}{3}} = \frac{1}{500}$

Step 2

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

$\ln (5^{\frac{y}{3}}) = \ln (\frac{1}{500})$

Step 3

Expand the left side.

Expand $\ln (5^{\frac{y}{3}})$ by moving $\frac{y}{3}$ outside the logarithm.

$\frac{y}{3} \ln (5) = \ln (\frac{1}{500})$

Combine $\frac{y}{3}$ and $\ln (5)$ .

$\frac{y \ln (5)}{3} = \ln (\frac{1}{500})$

Step 4

Multiply both sides of the equation by $\frac{3}{\ln (5)}$ .

$\frac{3}{\ln (5)} \cdot \frac{y \ln (5)}{3} = \frac{3}{\ln (5)} \ln (\frac{1}{500})$

Step 5

Simplify both sides of the equation.

Simplify the left side.

Simplify $\frac{3}{\ln (5)} \cdot \frac{y \ln (5)}{3}$ .

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3}{\ln (5)} \cdot \frac{y \ln (5)}{3} = \frac{3}{\ln (5)} \ln (\frac{1}{500})$

Rewrite the expression.

$\frac{1}{\ln (5)} (y \ln (5)) = \frac{3}{\ln (5)} \ln (\frac{1}{500})$

Cancel the common factor of $\ln (5)$ .

Factor $\ln (5)$ out of $y \ln (5)$ .

$\frac{1}{\ln (5)} (\ln (5) y) = \frac{3}{\ln (5)} \ln (\frac{1}{500})$

Cancel the common factor.

$\frac{1}{\ln (5)} (\ln (5) y) = \frac{3}{\ln (5)} \ln (\frac{1}{500})$

Rewrite the expression.

$y = \frac{3}{\ln (5)} \ln (\frac{1}{500})$

Simplify the right side.

Combine $\frac{3}{\ln (5)}$ and $\ln (\frac{1}{500})$ .

$y = \frac{3 \ln (\frac{1}{500})}{\ln (5)}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$y = \frac{3 \ln (\frac{1}{500})}{\ln (5)}$

Decimal Form:

$y = - 11.58405934 \dots$

Evaluate 500*5^(y/3)=1

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