Basic Math Examples

$4 \cdot 3^{x} = 5^{2 x - 1}$

Step 1

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

$\ln (4 \cdot 3^{x}) = \ln (5^{2 x - 1})$

Step 2

Expand the left side.

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Step 2.1

Rewrite $\ln (4 \cdot 3^{x})$ as $\ln (4) + \ln (3^{x})$ .

$\ln (4) + \ln (3^{x}) = \ln (5^{2 x - 1})$

Step 2.2

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

$\ln (4) + x \ln (3) = \ln (5^{2 x - 1})$

Step 3

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

$\ln (4) + x \ln (3) = (2 x - 1) \ln (5)$

Step 4

Simplify the right side.

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Step 4.1

Simplify $(2 x - 1) \ln (5)$ .

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Step 4.1.1

Apply the distributive property.

$\ln (4) + x \ln (3) = 2 x \ln (5) - 1 \ln (5)$

Step 4.1.2

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

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

Step 5

Reorder $\ln (4)$ and $x \ln (3)$ .

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

Step 6

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

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

Step 7

Use the product property of logarithms, $\log_{b} (x) + \log_{b} (y) = \log_{b} (x y)$ .

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

Step 8

Multiply $4$ by $5$ .

$x \ln (3) - 2 x \ln (5) + \ln (20) = 0$

Step 9

Subtract $\ln (20)$ from both sides of the equation.

$x \ln (3) - 2 x \ln (5) = - \ln (20)$

Step 10

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

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Step 10.1

Factor $x$ out of $x \ln (3)$ .

$x (\ln (3)) - 2 x \ln (5) = - \ln (20)$

Step 10.2

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

$x (\ln (3)) + x (- 2 \ln (5)) = - \ln (20)$

Step 10.3

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

$x (\ln (3) - 2 \ln (5)) = - \ln (20)$

Step 11

Divide each term in $x (\ln (3) - 2 \ln (5)) = - \ln (20)$ by $\ln (3) - 2 \ln (5)$ and simplify.

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Step 11.1

Divide each term in $x (\ln (3) - 2 \ln (5)) = - \ln (20)$ by $\ln (3) - 2 \ln (5)$ .

$\frac{x (\ln (3) - 2 \ln (5))}{\ln (3) - 2 \ln (5)} = \frac{- \ln (20)}{\ln (3) - 2 \ln (5)}$

Step 11.2

Simplify the left side.

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Step 11.2.1

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

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Step 11.2.1.1

Cancel the common factor.

$\frac{x (\ln (3) - 2 \ln (5))}{\ln (3) - 2 \ln (5)} = \frac{- \ln (20)}{\ln (3) - 2 \ln (5)}$

Step 11.2.1.2

Divide $x$ by $1$ .

$x = \frac{- \ln (20)}{\ln (3) - 2 \ln (5)}$

Step 11.3

Simplify the right side.

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Step 11.3.1

Move the negative in front of the fraction.

$x = - \frac{\ln (20)}{\ln (3) - 2 \ln (5)}$

Step 12

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{\ln (20)}{\ln (3) - 2 \ln (5)}$

Decimal Form:

$x = 1.41290562 \dots$