Evaluate Using the Given Value 4(5)^(x-2)=312 , 500

$4 {(5)}^{x - 2} = 312$ , $500$

Step 1

Divide each term in $(4 \cdot 5^{x - 2}) = 312$ by $4$ and simplify.

Divide each term in $(4 \cdot 5^{x - 2}) = 312$ by $4$ .

$\frac{(4 \cdot 5^{x - 2})}{4} = \frac{312}{4}$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 \cdot 5^{x - 2}}{4} = \frac{312}{4}$

Divide $5^{x - 2}$ by $1$ .

$5^{x - 2} = \frac{312}{4}$

Simplify the right side.

Divide $312$ by $4$ .

$5^{x - 2} = 78$

Step 2

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

$\ln (5^{x - 2}) = \ln (78)$

Step 3

Expand $\ln (5^{x - 2})$ by moving $x - 2$ outside the logarithm.

$(x - 2) \ln (5) = \ln (78)$

Step 4

Simplify the left side.

Apply the distributive property.

$x \ln (5) - 2 \ln (5) = \ln (78)$

Step 5

Move all the terms containing a logarithm to the left side of the equation.

$x \ln (5) - 2 \ln (5) - \ln (78) = 0$

Step 6

Move all terms not containing $x$ to the right side of the equation.

Add $2 \ln (5)$ to both sides of the equation.

$x \ln (5) - \ln (78) = 2 \ln (5)$

Add $\ln (78)$ to both sides of the equation.

$x \ln (5) = 2 \ln (5) + \ln (78)$

Step 7

Divide each term in $x \ln (5) = 2 \ln (5) + \ln (78)$ by $\ln (5)$ and simplify.

Divide each term in $x \ln (5) = 2 \ln (5) + \ln (78)$ by $\ln (5)$ .

$\frac{x \ln (5)}{\ln (5)} = \frac{2 \ln (5)}{\ln (5)} + \frac{\ln (78)}{\ln (5)}$

Simplify the left side.

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

Cancel the common factor.

$\frac{x \ln (5)}{\ln (5)} = \frac{2 \ln (5)}{\ln (5)} + \frac{\ln (78)}{\ln (5)}$

Divide $x$ by $1$ .

$x = \frac{2 \ln (5)}{\ln (5)} + \frac{\ln (78)}{\ln (5)}$

Simplify the right side.

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

Cancel the common factor.

$x = \frac{2 \ln (5)}{\ln (5)} + \frac{\ln (78)}{\ln (5)}$

Divide $2$ by $1$ .

$x = 2 + \frac{\ln (78)}{\ln (5)}$

Step 8

Replace the variable $x$ with $2 + \frac{\ln (78)}{\ln (5)}$ in the expression.

$500$

Evaluate Using the Given Value 4(5)^(x-2)=312 , 500

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