Algebra Examples

$\log_{3} (x) \cdot \log_{2} (3) = 5$

Step 1

Divide each term in $\log_{3} (x) \log_{2} (3) = 5$ by $\log_{2} (3)$ and simplify.

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Step 1.1

Divide each term in $\log_{3} (x) \log_{2} (3) = 5$ by $\log_{2} (3)$ .

$\frac{\log_{3} (x) \log_{2} (3)}{\log_{2} (3)} = \frac{5}{\log_{2} (3)}$

Step 1.2

Simplify the left side.

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Step 1.2.1

Cancel the common factor of $\log_{2} (3)$ .

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Step 1.2.1.1

Cancel the common factor.

$\frac{\log_{3} (x) \log_{2} (3)}{\log_{2} (3)} = \frac{5}{\log_{2} (3)}$

Step 1.2.1.2

Divide $\log_{3} (x)$ by $1$ .

$\log_{3} (x) = \frac{5}{\log_{2} (3)}$

Step 2

Rewrite $\log_{3} (x) = \frac{5}{\log_{2} (3)}$ in exponential form using the definition of a logarithm. If $x$ and $b$ are positive real numbers and $b \neq 1$ , then $\log_{b} (x) = y$ is equivalent to $b^{y} = x$ .

$3^{\frac{5}{\log_{2} (3)}} = x$

Step 3

Rewrite the equation as $x = 3^{\frac{5}{\log_{2} (3)}}$ .

$x = 3^{\frac{5}{\log_{2} (3)}}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$x = 3^{\frac{5}{\log_{2} (3)}}$

Decimal Form:

$x = 32$