Basic Math Examples

$5^{y + 1} - 5^{y} = 8$ fory

Step 1

Factor out $5^{y}$ from the expression.

$5^{y} (5 - 1)$

Step 2

Subtract $1$ from $5$ .

$5^{y} \cdot 4 = 8$

Step 3

Move $4$ to the left of $5^{y}$ .

$4 \cdot 5^{y} = 8$

Step 4

Divide each term in $4 \cdot 5^{y} = 8$ by $4$ and simplify.

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Step 4.1

Divide each term in $4 \cdot 5^{y} = 8$ by $4$ .

$\frac{4 \cdot 5^{y}}{4} = \frac{8}{4}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $4$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{4 \cdot 5^{y}}{4} = \frac{8}{4}$

Step 4.2.1.2

Divide $5^{y}$ by $1$ .

$5^{y} = \frac{8}{4}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Divide $8$ by $4$ .

$5^{y} = 2$

Step 5

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

$\ln (5^{y}) = \ln (2)$

Step 6

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

$y \ln (5) = \ln (2)$

Step 7

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

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Step 7.1

Divide each term in $y \ln (5) = \ln (2)$ by $\ln (5)$ .

$\frac{y \ln (5)}{\ln (5)} = \frac{\ln (2)}{\ln (5)}$

Step 7.2

Simplify the left side.

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Step 7.2.1

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

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Step 7.2.1.1

Cancel the common factor.

$\frac{y \ln (5)}{\ln (5)} = \frac{\ln (2)}{\ln (5)}$

Step 7.2.1.2

Divide $y$ by $1$ .

$y = \frac{\ln (2)}{\ln (5)}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$y = \frac{\ln (2)}{\ln (5)}$

Decimal Form:

$y = 0.43067655 \dots$