Algebra Examples

$\frac{\log_{\sqrt{2}} (\sqrt{5})}{\log_{\sqrt{2}} (\sqrt{10})}$

Step 1

Logarithm base $\sqrt{2}$ of $\sqrt{5}$ is $2.32192809$ .

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Step 1.1

Rewrite as an equation.

$\frac{\log_{\sqrt{2}} (\sqrt{5}) = x}{\log_{\sqrt{2}} (\sqrt{10})}$

Step 1.2

Rewrite $\log_{\sqrt{2}} (\sqrt{5}) = x$ in exponential form using the definition of a logarithm. If $x$ and $b$ are positive real numbers and $b$ does not equal $1$ , then $\log_{b} (x) = y$ is equivalent to $b^{y} = x$ .

$\frac{{\sqrt{2}}^{x} = \sqrt{5}}{\log_{\sqrt{2}} (\sqrt{10})}$

Step 1.3

Create equivalent expressions in the equation that all have equal bases.

$\frac{{\sqrt{2}}^{x} = {\sqrt{2}}^{2.32192809}}{\log_{\sqrt{2}} (\sqrt{10})}$

Step 1.4

Since the bases are the same, the two expressions are only equal if the exponents are also equal.

$\frac{x = 2.32192809}{\log_{\sqrt{2}} (\sqrt{10})}$

Step 1.5

The variable $x$ is equal to $2.32192809$ .

$\frac{2.32192809}{\log_{\sqrt{2}} (\sqrt{10})}$

Step 2

Logarithm base $\sqrt{2}$ of $\sqrt{10}$ is $3.32192809$ .

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Step 2.1

Rewrite as an equation.

$\frac{2.32192809}{\log_{\sqrt{2}} (\sqrt{10}) = x}$

Step 2.2

Rewrite $\log_{\sqrt{2}} (\sqrt{10}) = x$ in exponential form using the definition of a logarithm. If $x$ and $b$ are positive real numbers and $b$ does not equal $1$ , then $\log_{b} (x) = y$ is equivalent to $b^{y} = x$ .

$\frac{2.32192809}{{\sqrt{2}}^{x} = \sqrt{10}}$

Step 2.3

Create equivalent expressions in the equation that all have equal bases.

$\frac{2.32192809}{{\sqrt{2}}^{x} = {\sqrt{2}}^{3.32192809}}$

Step 2.4

Since the bases are the same, the two expressions are only equal if the exponents are also equal.

$\frac{2.32192809}{x = 3.32192809}$

Step 2.5

The variable $x$ is equal to $3.32192809$ .

$\frac{2.32192809}{3.32192809}$

Step 3

Divide $2.32192809$ by $3.32192809$ .

$0.69897$