Algebra Examples

$4^{y} < 5^{2 y + 1}$

Step 1

Take the log of both sides of the inequality.

$\ln (4^{y}) < \ln (5^{2 y + 1})$

Step 2

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

$y \ln (4) < \ln (5^{2 y + 1})$

Step 3

Expand $\ln (5^{2 y + 1})$ by moving $2 y + 1$ outside the logarithm.

$y \ln (4) < (2 y + 1) \ln (5)$

Step 4

Solve the inequality for $y$ .

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Step 4.1

Simplify $(2 y + 1) \ln (5)$ .

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Step 4.1.1

Apply the distributive property.

$y \ln (4) < 2 y \ln (5) + 1 \ln (5)$

Step 4.1.2

Multiply $\ln (5)$ by $1$ .

$y \ln (4) < 2 y \ln (5) + \ln (5)$

Step 4.2

Subtract $2 y \ln (5)$ from both sides of the inequality.

$y \ln (4) - 2 y \ln (5) < \ln (5)$

Step 4.3

Factor $y$ out of $y \ln (4) - 2 y \ln (5)$ .

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Step 4.3.1

Factor $y$ out of $y \ln (4)$ .

$y (\ln (4)) - 2 y \ln (5) < \ln (5)$

Step 4.3.2

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

$y (\ln (4)) + y (- 2 \ln (5)) < \ln (5)$

Step 4.3.3

Factor $y$ out of $y (\ln (4)) + y (- 2 \ln (5))$ .

$y (\ln (4) - 2 \ln (5)) < \ln (5)$

Step 4.4

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

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Step 4.4.1

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

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

Step 4.4.2

Simplify the left side.

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Step 4.4.2.1

Cancel the common factor of $\ln (4) - 2 \ln (5)$ .

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Step 4.4.2.1.1

Cancel the common factor.

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

Step 4.4.2.1.2

Divide $y$ by $1$ .

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

Step 5

The solution consists of all of the true intervals.

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

Step 6

The result can be shown in multiple forms.

Inequality Form:

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

Interval Notation:

$(\frac{\ln (5)}{\ln (4) - 2 \ln (5)}, \infty)$

Step 7