Algebra Examples

$\log_{x} (100) + \log_{x} (1000) = 5$

Step 1

Simplify the left side.

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Step 1.1

Use the product property of logarithms, $\log_{b} (x) + \log_{b} (y) = \log_{b} (x y)$ .

$\log_{x} (100 \cdot 1000) = 5$

Step 1.2

Multiply $100$ by $1000$ .

$\log_{x} (100000) = 5$

Step 2

Rewrite $\log_{x} (100000) = 5$ 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 $b^{y} = x$ .

$x^{5} = 100000$

Step 3

Solve for $x$ .

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Step 3.1

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

$x = \sqrt[5]{100000}$

Step 3.2

Simplify $\sqrt[5]{100000}$ .

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Step 3.2.1

Rewrite $100000$ as $10^{5}$ .

$x = \sqrt[5]{10^{5}}$

Step 3.2.2

Pull terms out from under the radical, assuming real numbers.

$x = 10$