Precalculus Examples

$45 {(\frac{2}{5})}^{x} = 10$

Step 1

Divide each term in $45 {(\frac{2}{5})}^{x} = 10$ by $45$ and simplify.

Tap for more steps...

Step 1.1

Divide each term in $45 {(\frac{2}{5})}^{x} = 10$ by $45$ .

$\frac{45 {(\frac{2}{5})}^{x}}{45} = \frac{10}{45}$

Step 1.2

Simplify the left side.

Tap for more steps...

Step 1.2.1

Cancel the common factor of $45$ .

Tap for more steps...

Step 1.2.1.1

Cancel the common factor.

$\frac{45 {(\frac{2}{5})}^{x}}{45} = \frac{10}{45}$

Step 1.2.1.2

Divide ${(\frac{2}{5})}^{x}$ by $1$ .

${(\frac{2}{5})}^{x} = \frac{10}{45}$

Step 1.2.2

Apply the product rule to $\frac{2}{5}$ .

$\frac{2^{x}}{5^{x}} = \frac{10}{45}$

Step 1.3

Simplify the right side.

Tap for more steps...

Step 1.3.1

Cancel the common factor of $10$ and $45$ .

Tap for more steps...

Step 1.3.1.1

Factor $5$ out of $10$ .

$\frac{2^{x}}{5^{x}} = \frac{5 (2)}{45}$

Step 1.3.1.2

Cancel the common factors.

Tap for more steps...

Step 1.3.1.2.1

Factor $5$ out of $45$ .

$\frac{2^{x}}{5^{x}} = \frac{5 \cdot 2}{5 \cdot 9}$

Step 1.3.1.2.2

Cancel the common factor.

$\frac{2^{x}}{5^{x}} = \frac{5 \cdot 2}{5 \cdot 9}$

Step 1.3.1.2.3

Rewrite the expression.

$\frac{2^{x}}{5^{x}} = \frac{2}{9}$

Step 2

Multiply both sides by $5^{x}$ .

$\frac{2^{x}}{5^{x}} \cdot 5^{x} = \frac{2}{9} \cdot 5^{x}$

Step 3

Simplify.

Tap for more steps...

Step 3.1

Simplify the left side.

Tap for more steps...

Step 3.1.1

Cancel the common factor of $5^{x}$ .

Tap for more steps...

Step 3.1.1.1

Cancel the common factor.

$\frac{2^{x}}{5^{x}} \cdot 5^{x} = \frac{2}{9} \cdot 5^{x}$

Step 3.1.1.2

Rewrite the expression.

$2^{x} = \frac{2}{9} \cdot 5^{x}$

Step 3.2

Simplify the right side.

Tap for more steps...

Step 3.2.1

Combine $\frac{2}{9}$ and $5^{x}$ .

$2^{x} = \frac{2 \cdot 5^{x}}{9}$

Step 4

Solve for $x$ .

Tap for more steps...

Step 4.1

Take the log of both sides of the equation.

$\ln (2^{x}) = \ln (\frac{2 \cdot 5^{x}}{9})$

Step 4.2

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

$x \ln (2) = \ln (\frac{2 \cdot 5^{x}}{9})$

Step 4.3

Rewrite $\ln (\frac{2 \cdot 5^{x}}{9})$ as $\ln (2 \cdot 5^{x}) - \ln (9)$ .

$x \ln (2) = \ln (2 \cdot 5^{x}) - \ln (9)$

Step 4.4

Rewrite $\ln (9)$ as $\ln (3^{2})$ .

$x \ln (2) = \ln (2 \cdot 5^{x}) - \ln (3^{2})$

Step 4.5

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

$x \ln (2) = \ln (2 \cdot 5^{x}) - (2 \ln (3))$

Step 4.6

Multiply $2$ by $- 1$ .

$x \ln (2) = \ln (2 \cdot 5^{x}) - 2 \ln (3)$

Step 4.7

Rewrite $\ln (2 \cdot 5^{x})$ as $\ln (2) + \ln (5^{x})$ .

$x \ln (2) = \ln (2) + \ln (5^{x}) - 2 \ln (3)$

Step 4.8

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

$x \ln (2) = \ln (2) + x \ln (5) - 2 \ln (3)$

Step 4.9

Solve the equation for $x$ .

Tap for more steps...

Step 4.9.1

Simplify the right side.

Tap for more steps...

Step 4.9.1.1

Simplify $\ln (2) + x \ln (5) - 2 \ln (3)$ .

Tap for more steps...

Step 4.9.1.1.1

Simplify each term.

Tap for more steps...

Step 4.9.1.1.1.1

Simplify $- 2 \ln (3)$ by moving $2$ inside the logarithm.

$x \ln (2) = \ln (2) + x \ln (5) - \ln (3^{2})$

Step 4.9.1.1.1.2

Raise $3$ to the power of $2$ .

$x \ln (2) = \ln (2) + x \ln (5) - \ln (9)$

Step 4.9.1.1.2

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$x \ln (2) = x \ln (5) + \ln (\frac{2}{9})$

Step 4.9.2

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

$x \ln (2) - x \ln (5) - \ln (\frac{2}{9}) = 0$

Step 4.9.3

Add $\ln (\frac{2}{9})$ to both sides of the equation.

$x \ln (2) - x \ln (5) = \ln (\frac{2}{9})$

Step 4.9.4

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

Tap for more steps...

Step 4.9.4.1

Factor $x$ out of $x \ln (2)$ .

$x (\ln (2)) - x \ln (5) = \ln (\frac{2}{9})$

Step 4.9.4.2

Factor $x$ out of $- x \ln (5)$ .

$x (\ln (2)) + x (- 1 \ln (5)) = \ln (\frac{2}{9})$

Step 4.9.4.3

Factor $x$ out of $x (\ln (2)) + x (- 1 \ln (5))$ .

$x (\ln (2) - 1 \ln (5)) = \ln (\frac{2}{9})$

Step 4.9.5

Rewrite $- 1 \ln (5)$ as $- \ln (5)$ .

$x (\ln (2) - \ln (5)) = \ln (\frac{2}{9})$

Step 4.9.6

Divide each term in $x (\ln (2) - \ln (5)) = \ln (\frac{2}{9})$ by $\ln (2) - \ln (5)$ and simplify.

Tap for more steps...

Step 4.9.6.1

Divide each term in $x (\ln (2) - \ln (5)) = \ln (\frac{2}{9})$ by $\ln (2) - \ln (5)$ .

$\frac{x (\ln (2) - \ln (5))}{\ln (2) - \ln (5)} = \frac{\ln (\frac{2}{9})}{\ln (2) - \ln (5)}$

Step 4.9.6.2

Simplify the left side.

Tap for more steps...

Step 4.9.6.2.1

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

Tap for more steps...

Step 4.9.6.2.1.1

Cancel the common factor.

$\frac{x (\ln (2) - \ln (5))}{\ln (2) - \ln (5)} = \frac{\ln (\frac{2}{9})}{\ln (2) - \ln (5)}$

Step 4.9.6.2.1.2

Divide $x$ by $1$ .

$x = \frac{\ln (\frac{2}{9})}{\ln (2) - \ln (5)}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = \frac{\ln (\frac{2}{9})}{\ln (2) - \ln (5)}$

Decimal Form:

$x = 1.64148489 \dots$