Basic Math Examples

$5^{y - 4} = 3^{2 y}$

Step 1

Take the log of both sides of the equation.

$\ln (5^{y - 4}) = \ln (3^{2 y})$

Step 2

Expand $\ln (5^{y - 4})$ by moving $y - 4$ outside the logarithm.

$(y - 4) \ln (5) = \ln (3^{2 y})$

Step 3

Expand $\ln (3^{2 y})$ by moving $2 y$ outside the logarithm.

$(y - 4) \ln (5) = 2 y \ln (3)$

Step 4

Solve the equation for $y$ .

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Step 4.1

Simplify $(y - 4) \ln (5)$ .

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Step 4.1.1

Rewrite.

$0 + 0 + (y - 4) \ln (5) = 2 y \ln (3)$

Step 4.1.2

Simplify by adding zeros.

$(y - 4) \ln (5) = 2 y \ln (3)$

Step 4.1.3

Apply the distributive property.

$y \ln (5) - 4 \ln (5) = 2 y \ln (3)$

Step 4.2

Subtract $2 y \ln (3)$ from both sides of the equation.

$y \ln (5) - 4 \ln (5) - 2 y \ln (3) = 0$

Step 4.3

Add $4 \ln (5)$ to both sides of the equation.

$y \ln (5) - 2 y \ln (3) = 4 \ln (5)$

Step 4.4

Factor $y$ out of $y \ln (5) - 2 y \ln (3)$ .

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Step 4.4.1

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

$y (\ln (5)) - 2 y \ln (3) = 4 \ln (5)$

Step 4.4.2

Factor $y$ out of $- 2 y \ln (3)$ .

$y (\ln (5)) + y (- 2 \ln (3)) = 4 \ln (5)$

Step 4.4.3

Factor $y$ out of $y (\ln (5)) + y (- 2 \ln (3))$ .

$y (\ln (5) - 2 \ln (3)) = 4 \ln (5)$

Step 4.5

Divide each term in $y (\ln (5) - 2 \ln (3)) = 4 \ln (5)$ by $\ln (5) - 2 \ln (3)$ and simplify.

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Step 4.5.1

Divide each term in $y (\ln (5) - 2 \ln (3)) = 4 \ln (5)$ by $\ln (5) - 2 \ln (3)$ .

$\frac{y (\ln (5) - 2 \ln (3))}{\ln (5) - 2 \ln (3)} = \frac{4 \ln (5)}{\ln (5) - 2 \ln (3)}$

Step 4.5.2

Simplify the left side.

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Step 4.5.2.1

Cancel the common factor of $\ln (5) - 2 \ln (3)$ .

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Step 4.5.2.1.1

Cancel the common factor.

$\frac{y (\ln (5) - 2 \ln (3))}{\ln (5) - 2 \ln (3)} = \frac{4 \ln (5)}{\ln (5) - 2 \ln (3)}$

Step 4.5.2.1.2

Divide $y$ by $1$ .

$y = \frac{4 \ln (5)}{\ln (5) - 2 \ln (3)}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$y = \frac{4 \ln (5)}{\ln (5) - 2 \ln (3)}$

Decimal Form:

$y = - 10.95253096 \dots$