Precalculus Examples

$5 \ln (x) = \ln (27) - \ln (3)$

Step 1

Simplify the left side.

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Step 1.1

Simplify $5 \ln (x)$ by moving $5$ inside the logarithm.

$\ln (x^{5}) = \ln (27) - \ln (3)$

Step 2

Simplify the right side.

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Step 2.1

Simplify $\ln (27) - \ln (3)$ .

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Step 2.1.1

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

$\ln (x^{5}) = \ln (\frac{27}{3})$

Step 2.1.2

Divide $27$ by $3$ .

$\ln (x^{5}) = \ln (9)$

Step 3

For the equation to be equal, the argument of the logarithms on both sides of the equation must be equal.

$x^{5} = 9$

Step 4

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \sqrt[5]{9}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = \sqrt[5]{9}$

Decimal Form:

$x = 1.55184557 \dots$