Solve for x 2.4^(3x+1)=9

${2.4}^{3 x + 1} = 9$

Step 1

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

$\ln ({2.4}^{3 x + 1}) = \ln (9)$

Step 2

Expand $\ln ({2.4}^{3 x + 1})$ by moving $3 x + 1$ outside the logarithm.

$(3 x + 1) \ln (2.4) = \ln (9)$

Step 3

Simplify the left side.

Simplify $(3 x + 1) \ln (2.4)$ .

Apply the distributive property.

$3 x \ln (2.4) + 1 \ln (2.4) = \ln (9)$

Multiply $\ln (2.4)$ by $1$ .

$3 x \ln (2.4) + \ln (2.4) = \ln (9)$

Step 4

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

$3 x \ln (2.4) + \ln (2.4) - \ln (9) = 0$

Step 5

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

$3 x \ln (2.4) + \ln (\frac{2.4}{9}) = 0$

Step 6

Divide $2.4$ by $9$ .

$3 x \ln (2.4) + \ln (0.2\overline{6}) = 0$

Step 7

Subtract $\ln (0.2\overline{6})$ from both sides of the equation.

$3 x \ln (2.4) = - \ln (0.2\overline{6})$

Step 8

Divide each term in $3 x \ln (2.4) = - \ln (0.2\overline{6})$ by $3 \ln (2.4)$ and simplify.

Divide each term in $3 x \ln (2.4) = - \ln (0.2\overline{6})$ by $3 \ln (2.4)$ .

$\frac{3 x \ln (2.4)}{3 \ln (2.4)} = \frac{- \ln (0.2\overline{6})}{3 \ln (2.4)}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 x \ln (2.4)}{3 \ln (2.4)} = \frac{- \ln (0.2\overline{6})}{3 \ln (2.4)}$

Rewrite the expression.

$\frac{x \ln (2.4)}{\ln (2.4)} = \frac{- \ln (0.2\overline{6})}{3 \ln (2.4)}$

Cancel the common factor of $\ln (2.4)$ .

Cancel the common factor.

$\frac{x \ln (2.4)}{\ln (2.4)} = \frac{- \ln (0.2\overline{6})}{3 \ln (2.4)}$

Divide $x$ by $1$ .

$x = \frac{- \ln (0.2\overline{6})}{3 \ln (2.4)}$

Simplify the right side.

Move the negative in front of the fraction.

$x = - \frac{\ln (0.2\overline{6})}{3 \ln (2.4)}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{\ln (0.2\overline{6})}{3 \ln (2.4)}$

Decimal Form:

$x = 0.50325643 \dots$

Solve for x 2.4^(3x+1)=9

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