Calculus Examples

$8^{x} = \frac{5^{x + 2}}{3}$

Step 1

Take the log of both sides of the equation.

$\ln (8^{x}) = \ln (\frac{5^{x + 2}}{3})$

Step 2

Expand $\ln (8^{x})$ by moving $x$ outside the logarithm.

$x \ln (8) = \ln (\frac{5^{x + 2}}{3})$

Step 3

Rewrite $\ln (\frac{5^{x + 2}}{3})$ as $\ln (5^{x + 2}) - \ln (3)$ .

$x \ln (8) = \ln (5^{x + 2}) - \ln (3)$

Step 4

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

$x \ln (8) = (x + 2) \ln (5) - \ln (3)$

Step 5

Solve the equation for $x$ .

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Step 5.1

Simplify the right side.

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Step 5.1.1

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

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Step 5.1.1.1

Simplify each term.

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Step 5.1.1.1.1

Apply the distributive property.

$x \ln (8) = x \ln (5) + 2 \ln (5) - \ln (3)$

Step 5.1.1.1.2

Simplify $2 \ln (5)$ by moving $2$ inside the logarithm.

$x \ln (8) = x \ln (5) + \ln (5^{2}) - \ln (3)$

Step 5.1.1.1.3

Raise $5$ to the power of $2$ .

$x \ln (8) = x \ln (5) + \ln (25) - \ln (3)$

Step 5.1.1.2

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

$x \ln (8) = x \ln (5) + \ln (\frac{25}{3})$

Step 5.2

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

$x \ln (8) - x \ln (5) - \ln (\frac{25}{3}) = 0$

Step 5.3

Add $\ln (\frac{25}{3})$ to both sides of the equation.

$x \ln (8) - x \ln (5) = \ln (\frac{25}{3})$

Step 5.4

Factor $x$ out of $x \ln (8) - x \ln (5)$ .

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Step 5.4.1

Factor $x$ out of $x \ln (8)$ .

$x (\ln (8)) - x \ln (5) = \ln (\frac{25}{3})$

Step 5.4.2

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

$x (\ln (8)) + x (- 1 \ln (5)) = \ln (\frac{25}{3})$

Step 5.4.3

Factor $x$ out of $x (\ln (8)) + x (- 1 \ln (5))$ .

$x (\ln (8) - 1 \ln (5)) = \ln (\frac{25}{3})$

Step 5.5

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

$x (\ln (8) - \ln (5)) = \ln (\frac{25}{3})$

Step 5.6

Divide each term in $x (\ln (8) - \ln (5)) = \ln (\frac{25}{3})$ by $\ln (8) - \ln (5)$ and simplify.

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Step 5.6.1

Divide each term in $x (\ln (8) - \ln (5)) = \ln (\frac{25}{3})$ by $\ln (8) - \ln (5)$ .

$\frac{x (\ln (8) - \ln (5))}{\ln (8) - \ln (5)} = \frac{\ln (\frac{25}{3})}{\ln (8) - \ln (5)}$

Step 5.6.2

Simplify the left side.

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Step 5.6.2.1

Cancel the common factor of $\ln (8) - \ln (5)$ .

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Step 5.6.2.1.1

Cancel the common factor.

$\frac{x (\ln (8) - \ln (5))}{\ln (8) - \ln (5)} = \frac{\ln (\frac{25}{3})}{\ln (8) - \ln (5)}$

Step 5.6.2.1.2

Divide $x$ by $1$ .

$x = \frac{\ln (\frac{25}{3})}{\ln (8) - \ln (5)}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$x = \frac{\ln (\frac{25}{3})}{\ln (8) - \ln (5)}$

Decimal Form:

$x = 4.51116417 \dots$