Algebra Examples

$5^{x - 3} = 4^{3 - x}$

Step 1

Take the log of both sides of the equation.

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

Step 2

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

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

Step 3

Expand $\ln (4^{3 - x})$ by moving $3 - x$ outside the logarithm.

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

Step 4

Solve the equation for $x$ .

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Step 4.1

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

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Step 4.1.1

Rewrite.

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

Step 4.1.2

Simplify by adding zeros.

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

Step 4.1.3

Apply the distributive property.

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

Step 4.2

Apply the distributive property.

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

Step 4.3

Add $x \ln (4)$ to both sides of the equation.

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

Step 4.4

Add $3 \ln (5)$ to both sides of the equation.

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

Step 4.5

Factor $x$ out of $x \ln (5) + x \ln (4)$ .

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Step 4.5.1

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

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

Step 4.5.2

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

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

Step 4.5.3

Factor $x$ out of $x (\ln (5)) + x (\ln (4))$ .

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

Step 4.6

Divide each term in $x (\ln (5) + \ln (4)) = 3 \ln (4) + 3 \ln (5)$ by $\ln (5) + \ln (4)$ and simplify.

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Step 4.6.1

Divide each term in $x (\ln (5) + \ln (4)) = 3 \ln (4) + 3 \ln (5)$ by $\ln (5) + \ln (4)$ .

$\frac{x (\ln (5) + \ln (4))}{\ln (5) + \ln (4)} = \frac{3 \ln (4)}{\ln (5) + \ln (4)} + \frac{3 \ln (5)}{\ln (5) + \ln (4)}$

Step 4.6.2

Simplify the left side.

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Step 4.6.2.1

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

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Step 4.6.2.1.1

Cancel the common factor.

$\frac{x (\ln (5) + \ln (4))}{\ln (5) + \ln (4)} = \frac{3 \ln (4)}{\ln (5) + \ln (4)} + \frac{3 \ln (5)}{\ln (5) + \ln (4)}$

Step 4.6.2.1.2

Divide $x$ by $1$ .

$x = \frac{3 \ln (4)}{\ln (5) + \ln (4)} + \frac{3 \ln (5)}{\ln (5) + \ln (4)}$

Step 4.6.3

Simplify the right side.

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Step 4.6.3.1

Combine the numerators over the common denominator.

$x = \frac{3 \ln (4) + 3 \ln (5)}{\ln (5) + \ln (4)}$

Step 4.6.3.2

Factor $3$ out of $3 \ln (4) + 3 \ln (5)$ .

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Step 4.6.3.2.1

Factor $3$ out of $3 \ln (4)$ .

$x = \frac{3 \ln (4) + 3 \ln (5)}{\ln (5) + \ln (4)}$

Step 4.6.3.2.2

Factor $3$ out of $3 \ln (5)$ .

$x = \frac{3 \ln (4) + 3 \ln (5)}{\ln (5) + \ln (4)}$

Step 4.6.3.2.3

Factor $3$ out of $3 \ln (4) + 3 \ln (5)$ .

$x = \frac{3 (\ln (4) + \ln (5))}{\ln (5) + \ln (4)}$

Step 4.6.3.3

Cancel the common factor of $\ln (4) + \ln (5)$ and $\ln (5) + \ln (4)$ .

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Step 4.6.3.3.1

Reorder terms.

$x = \frac{3 (\ln (4) + \ln (5))}{\ln (4) + \ln (5)}$

Step 4.6.3.3.2

Cancel the common factor.

$x = \frac{3 (\ln (4) + \ln (5))}{\ln (4) + \ln (5)}$

Step 4.6.3.3.3

Divide $3$ by $1$ .

$x = 3$