Precalculus Examples

$5^{5 - 3 x} = 26$

Step 1

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

$\ln (5^{5 - 3 x}) = \ln (26)$

Step 2

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

$(5 - 3 x) \ln (5) = \ln (26)$

Step 3

Simplify the left side.

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Step 3.1

Apply the distributive property.

$5 \ln (5) - 3 x \ln (5) = \ln (26)$

Step 4

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

$- 3 x \ln (5) + 5 \ln (5) = \ln (26)$

Step 5

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

$- 3 x \ln (5) + 5 \ln (5) - \ln (26) = 0$

Step 6

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

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Step 6.1

Subtract $5 \ln (5)$ from both sides of the equation.

$- 3 x \ln (5) - \ln (26) = - 5 \ln (5)$

Step 6.2

Add $\ln (26)$ to both sides of the equation.

$- 3 x \ln (5) = - 5 \ln (5) + \ln (26)$

Step 7

Divide each term in $- 3 x \ln (5) = - 5 \ln (5) + \ln (26)$ by $- 3 \ln (5)$ and simplify.

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Step 7.1

Divide each term in $- 3 x \ln (5) = - 5 \ln (5) + \ln (26)$ by $- 3 \ln (5)$ .

$\frac{- 3 x \ln (5)}{- 3 \ln (5)} = \frac{- 5 \ln (5)}{- 3 \ln (5)} + \frac{\ln (26)}{- 3 \ln (5)}$

Step 7.2

Simplify the left side.

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Step 7.2.1

Cancel the common factor of $- 3$ .

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Step 7.2.1.1

Cancel the common factor.

$\frac{- 3 x \ln (5)}{- 3 \ln (5)} = \frac{- 5 \ln (5)}{- 3 \ln (5)} + \frac{\ln (26)}{- 3 \ln (5)}$

Step 7.2.1.2

Rewrite the expression.

$\frac{x \ln (5)}{\ln (5)} = \frac{- 5 \ln (5)}{- 3 \ln (5)} + \frac{\ln (26)}{- 3 \ln (5)}$

Step 7.2.2

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

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Step 7.2.2.1

Cancel the common factor.

$\frac{x \ln (5)}{\ln (5)} = \frac{- 5 \ln (5)}{- 3 \ln (5)} + \frac{\ln (26)}{- 3 \ln (5)}$

Step 7.2.2.2

Divide $x$ by $1$ .

$x = \frac{- 5 \ln (5)}{- 3 \ln (5)} + \frac{\ln (26)}{- 3 \ln (5)}$

Step 7.3

Simplify the right side.

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Step 7.3.1

Simplify each term.

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Step 7.3.1.1

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

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Step 7.3.1.1.1

Cancel the common factor.

$x = \frac{- 5 \ln (5)}{- 3 \ln (5)} + \frac{\ln (26)}{- 3 \ln (5)}$

Step 7.3.1.1.2

Rewrite the expression.

$x = \frac{- 5}{- 3} + \frac{\ln (26)}{- 3 \ln (5)}$

Step 7.3.1.2

Dividing two negative values results in a positive value.

$x = \frac{5}{3} + \frac{\ln (26)}{- 3 \ln (5)}$

Step 7.3.1.3

Move the negative in front of the fraction.

$x = \frac{5}{3} - \frac{\ln (26)}{3 \ln (5)}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$x = \frac{5}{3} - \frac{\ln (26)}{3 \ln (5)}$

Decimal Form:

$x = 0.99187693 \dots$