Precalculus Examples

$5 (3^{x}) = 7$

Step 1

Divide each term in $5 \cdot 3^{x} = 7$ by $5$ and simplify.

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Step 1.1

Divide each term in $5 \cdot 3^{x} = 7$ by $5$ .

$\frac{5 \cdot 3^{x}}{5} = \frac{7}{5}$

Step 1.2

Simplify the left side.

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Step 1.2.1

Cancel the common factor of $5$ .

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Step 1.2.1.1

Cancel the common factor.

$\frac{5 \cdot 3^{x}}{5} = \frac{7}{5}$

Step 1.2.1.2

Divide $3^{x}$ by $1$ .

$3^{x} = \frac{7}{5}$

Step 2

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

$\ln (3^{x}) = \ln (\frac{7}{5})$

Step 3

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

$x \ln (3) = \ln (\frac{7}{5})$

Step 4

Divide each term in $x \ln (3) = \ln (\frac{7}{5})$ by $\ln (3)$ and simplify.

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Step 4.1

Divide each term in $x \ln (3) = \ln (\frac{7}{5})$ by $\ln (3)$ .

$\frac{x \ln (3)}{\ln (3)} = \frac{\ln (\frac{7}{5})}{\ln (3)}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $\ln (3)$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{x \ln (3)}{\ln (3)} = \frac{\ln (\frac{7}{5})}{\ln (3)}$

Step 4.2.1.2

Divide $x$ by $1$ .

$x = \frac{\ln (\frac{7}{5})}{\ln (3)}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = \frac{\ln (\frac{7}{5})}{\ln (3)}$

Decimal Form:

$x = 0.30627022 \dots$