Algebra Examples

$8 (2^{x}) = 7^{5 - 3 x}$

Step 1

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

$\ln (8 \cdot 2^{x}) = \ln (7^{5 - 3 x})$

Step 2

Expand the left side.

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Step 2.1

Rewrite $\ln (8 \cdot 2^{x})$ as $\ln (8) + \ln (2^{x})$ .

$\ln (8) + \ln (2^{x}) = \ln (7^{5 - 3 x})$

Step 2.2

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

$\ln (8) + x \ln (2) = \ln (7^{5 - 3 x})$

Step 3

Expand $\ln (7^{5 - 3 x})$ by moving $5 - 3 x$ outside the logarithm.

$\ln (8) + x \ln (2) = (5 - 3 x) \ln (7)$

Step 4

Simplify the right side.

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Step 4.1

Apply the distributive property.

$\ln (8) + x \ln (2) = 5 \ln (7) - 3 x \ln (7)$

Step 5

Reorder $\ln (8)$ and $x \ln (2)$ .

$x \ln (2) + \ln (8) = 5 \ln (7) - 3 x \ln (7)$

Step 6

Reorder $5 \ln (7)$ and $- 3 x \ln (7)$ .

$x \ln (2) + \ln (8) = - 3 x \ln (7) + 5 \ln (7)$

Step 7

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

$x \ln (2) + \ln (8) + 3 x \ln (7) - 5 \ln (7) = 0$

Step 8

Move all terms not containing $x$ to the right side of the equation.

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Step 8.1

Subtract $\ln (8)$ from both sides of the equation.

$x \ln (2) + 3 x \ln (7) - 5 \ln (7) = - \ln (8)$

Step 8.2

Add $5 \ln (7)$ to both sides of the equation.

$x \ln (2) + 3 x \ln (7) = - \ln (8) + 5 \ln (7)$

Step 9

Factor $x$ out of $x \ln (2) + 3 x \ln (7)$ .

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Step 9.1

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

$x (\ln (2)) + 3 x \ln (7) = - \ln (8) + 5 \ln (7)$

Step 9.2

Factor $x$ out of $3 x \ln (7)$ .

$x (\ln (2)) + x (3 \ln (7)) = - \ln (8) + 5 \ln (7)$

Step 9.3

Factor $x$ out of $x (\ln (2)) + x (3 \ln (7))$ .

$x (\ln (2) + 3 \ln (7)) = - \ln (8) + 5 \ln (7)$

Step 10

Divide each term in $x (\ln (2) + 3 \ln (7)) = - \ln (8) + 5 \ln (7)$ by $\ln (2) + 3 \ln (7)$ and simplify.

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Step 10.1

Divide each term in $x (\ln (2) + 3 \ln (7)) = - \ln (8) + 5 \ln (7)$ by $\ln (2) + 3 \ln (7)$ .

$\frac{x (\ln (2) + 3 \ln (7))}{\ln (2) + 3 \ln (7)} = \frac{- \ln (8)}{\ln (2) + 3 \ln (7)} + \frac{5 \ln (7)}{\ln (2) + 3 \ln (7)}$

Step 10.2

Simplify the left side.

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Step 10.2.1

Cancel the common factor of $\ln (2) + 3 \ln (7)$ .

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Step 10.2.1.1

Cancel the common factor.

$\frac{x (\ln (2) + 3 \ln (7))}{\ln (2) + 3 \ln (7)} = \frac{- \ln (8)}{\ln (2) + 3 \ln (7)} + \frac{5 \ln (7)}{\ln (2) + 3 \ln (7)}$

Step 10.2.1.2

Divide $x$ by $1$ .

$x = \frac{- \ln (8)}{\ln (2) + 3 \ln (7)} + \frac{5 \ln (7)}{\ln (2) + 3 \ln (7)}$

Step 10.3

Simplify the right side.

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Step 10.3.1

Combine the numerators over the common denominator.

$x = \frac{- \ln (8) + 5 \ln (7)}{\ln (2) + 3 \ln (7)}$

Step 10.3.2

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

$x = \frac{- \ln (8) + 5 \ln (7)}{\ln (2) + 3 \ln (7)}$

Step 10.3.3

Factor $- 1$ out of $5 \ln (7)$ .

$x = \frac{- \ln (8) - (- 5 \ln (7))}{\ln (2) + 3 \ln (7)}$

Step 10.3.4

Factor $- 1$ out of $- \ln (8) - (- 5 \ln (7))$ .

$x = \frac{- (\ln (8) - 5 \ln (7))}{\ln (2) + 3 \ln (7)}$

Step 10.3.5

Simplify the expression.

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Step 10.3.5.1

Rewrite $- (\ln (8) - 5 \ln (7))$ as $- 1 (\ln (8) - 5 \ln (7))$ .

$x = \frac{- 1 (\ln (8) - 5 \ln (7))}{\ln (2) + 3 \ln (7)}$

Step 10.3.5.2

Move the negative in front of the fraction.

$x = - \frac{\ln (8) - 5 \ln (7)}{\ln (2) + 3 \ln (7)}$

Step 11

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{\ln (8) - 5 \ln (7)}{\ln (2) + 3 \ln (7)}$

Decimal Form:

$x = 1.17137537 \dots$