Precalculus Examples

$5^{3 x + 4} = 29$

Step 1

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

$\ln (5^{3 x + 4}) = \ln (29)$

Step 2

Expand $\ln (5^{3 x + 4})$ by moving $3 x + 4$ outside the logarithm.

$(3 x + 4) \ln (5) = \ln (29)$

Step 3

Simplify the left side.

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Step 3.1

Apply the distributive property.

$3 x \ln (5) + 4 \ln (5) = \ln (29)$

Step 4

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

$3 x \ln (5) + 4 \ln (5) - \ln (29) = 0$

Step 5

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

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Step 5.1

Subtract $4 \ln (5)$ from both sides of the equation.

$3 x \ln (5) - \ln (29) = - 4 \ln (5)$

Step 5.2

Add $\ln (29)$ to both sides of the equation.

$3 x \ln (5) = - 4 \ln (5) + \ln (29)$

Step 6

Divide each term in $3 x \ln (5) = - 4 \ln (5) + \ln (29)$ by $3 \ln (5)$ and simplify.

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Step 6.1

Divide each term in $3 x \ln (5) = - 4 \ln (5) + \ln (29)$ by $3 \ln (5)$ .

$\frac{3 x \ln (5)}{3 \ln (5)} = \frac{- 4 \ln (5)}{3 \ln (5)} + \frac{\ln (29)}{3 \ln (5)}$

Step 6.2

Simplify the left side.

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Step 6.2.1

Cancel the common factor of $3$ .

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Step 6.2.1.1

Cancel the common factor.

$\frac{3 x \ln (5)}{3 \ln (5)} = \frac{- 4 \ln (5)}{3 \ln (5)} + \frac{\ln (29)}{3 \ln (5)}$

Step 6.2.1.2

Rewrite the expression.

$\frac{x \ln (5)}{\ln (5)} = \frac{- 4 \ln (5)}{3 \ln (5)} + \frac{\ln (29)}{3 \ln (5)}$

Step 6.2.2

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

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Step 6.2.2.1

Cancel the common factor.

$\frac{x \ln (5)}{\ln (5)} = \frac{- 4 \ln (5)}{3 \ln (5)} + \frac{\ln (29)}{3 \ln (5)}$

Step 6.2.2.2

Divide $x$ by $1$ .

$x = \frac{- 4 \ln (5)}{3 \ln (5)} + \frac{\ln (29)}{3 \ln (5)}$

Step 6.3

Simplify the right side.

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Step 6.3.1

Simplify each term.

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Step 6.3.1.1

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

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Step 6.3.1.1.1

Cancel the common factor.

$x = \frac{- 4 \ln (5)}{3 \ln (5)} + \frac{\ln (29)}{3 \ln (5)}$

Step 6.3.1.1.2

Rewrite the expression.

$x = \frac{- 4}{3} + \frac{\ln (29)}{3 \ln (5)}$

Step 6.3.1.2

Move the negative in front of the fraction.

$x = - \frac{4}{3} + \frac{\ln (29)}{3 \ln (5)}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{4}{3} + \frac{\ln (29)}{3 \ln (5)}$

Decimal Form:

$x = - 0.63592715 \dots$