Precalculus Examples

$5 {(1.2)}^{3 x - 2} + 1 = 7$

Step 1

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

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Step 1.1

Subtract $1$ from both sides of the equation.

$5 \cdot {1.2}^{3 x - 2} = 7 - 1$

Step 1.2

Subtract $1$ from $7$ .

$5 \cdot {1.2}^{3 x - 2} = 6$

Step 2

Divide each term in $5 \cdot {1.2}^{3 x - 2} = 6$ by $5$ and simplify.

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Step 2.1

Divide each term in $5 \cdot {1.2}^{3 x - 2} = 6$ by $5$ .

$\frac{5 \cdot {1.2}^{3 x - 2}}{5} = \frac{6}{5}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $5$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{5 \cdot {1.2}^{3 x - 2}}{5} = \frac{6}{5}$

Step 2.2.1.2

Divide ${1.2}^{3 x - 2}$ by $1$ .

${1.2}^{3 x - 2} = \frac{6}{5}$

Step 3

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

$\ln ({1.2}^{3 x - 2}) = \ln (\frac{6}{5})$

Step 4

Expand $\ln ({1.2}^{3 x - 2})$ by moving $3 x - 2$ outside the logarithm.

$(3 x - 2) \ln (1.2) = \ln (\frac{6}{5})$

Step 5

Simplify the left side.

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Step 5.1

Apply the distributive property.

$3 x \ln (1.2) - 2 \ln (1.2) = \ln (\frac{6}{5})$

Step 6

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

$3 x \ln (1.2) - 2 \ln (1.2) - \ln (\frac{6}{5}) = 0$

Step 7

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

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Step 7.1

Add $2 \ln (1.2)$ to both sides of the equation.

$3 x \ln (1.2) - \ln (\frac{6}{5}) = 2 \ln (1.2)$

Step 7.2

Add $\ln (\frac{6}{5})$ to both sides of the equation.

$3 x \ln (1.2) = 2 \ln (1.2) + \ln (\frac{6}{5})$

Step 8

Divide each term in $3 x \ln (1.2) = 2 \ln (1.2) + \ln (\frac{6}{5})$ by $3 \ln (1.2)$ and simplify.

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Step 8.1

Divide each term in $3 x \ln (1.2) = 2 \ln (1.2) + \ln (\frac{6}{5})$ by $3 \ln (1.2)$ .

$\frac{3 x \ln (1.2)}{3 \ln (1.2)} = \frac{2 \ln (1.2)}{3 \ln (1.2)} + \frac{\ln (\frac{6}{5})}{3 \ln (1.2)}$

Step 8.2

Simplify the left side.

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Step 8.2.1

Cancel the common factor of $3$ .

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Step 8.2.1.1

Cancel the common factor.

$\frac{3 x \ln (1.2)}{3 \ln (1.2)} = \frac{2 \ln (1.2)}{3 \ln (1.2)} + \frac{\ln (\frac{6}{5})}{3 \ln (1.2)}$

Step 8.2.1.2

Rewrite the expression.

$\frac{x \ln (1.2)}{\ln (1.2)} = \frac{2 \ln (1.2)}{3 \ln (1.2)} + \frac{\ln (\frac{6}{5})}{3 \ln (1.2)}$

Step 8.2.2

Cancel the common factor of $\ln (1.2)$ .

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Step 8.2.2.1

Cancel the common factor.

$\frac{x \ln (1.2)}{\ln (1.2)} = \frac{2 \ln (1.2)}{3 \ln (1.2)} + \frac{\ln (\frac{6}{5})}{3 \ln (1.2)}$

Step 8.2.2.2

Divide $x$ by $1$ .

$x = \frac{2 \ln (1.2)}{3 \ln (1.2)} + \frac{\ln (\frac{6}{5})}{3 \ln (1.2)}$

Step 8.3

Simplify the right side.

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Step 8.3.1

Cancel the common factor of $\ln (1.2)$ .

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Step 8.3.1.1

Cancel the common factor.

$x = \frac{2 \ln (1.2)}{3 \ln (1.2)} + \frac{\ln (\frac{6}{5})}{3 \ln (1.2)}$

Step 8.3.1.2

Rewrite the expression.

$x = \frac{2}{3} + \frac{\ln (\frac{6}{5})}{3 \ln (1.2)}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$x = \frac{2}{3} + \frac{\ln (\frac{6}{5})}{3 \ln (1.2)}$

Decimal Form:

$x = 1$