Algebra Examples

$\log (x) - \log (10) = 14$

Step 1

Simplify the left side.

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Step 1.1

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

$\log (\frac{x}{10}) = 14$

Step 1.2

Rewrite $\frac{x}{10}$ as $x \cdot 10^{- 1}$ .

$\log (x \cdot 10^{- 1}) = 14$

Step 1.3

Rewrite $\log (x \cdot 10^{- 1})$ as $\log (x) + \log (10^{- 1})$ .

$\log (x) + \log (10^{- 1}) = 14$

Step 1.4

Use logarithm rules to move $- 1$ out of the exponent.

$\log (x) - \log (10) = 14$

Step 1.5

Logarithm base $10$ of $10$ is $1$ .

$\log (x) - 1 \cdot 1 = 14$

Step 1.6

Multiply $- 1$ by $1$ .

$\log (x) - 1 = 14$

Step 2

Move all terms not containing $\log (x)$ to the right side of the equation.

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Step 2.1

Add $1$ to both sides of the equation.

$\log (x) = 14 + 1$

Step 2.2

Add $14$ and $1$ .

$\log (x) = 15$

Step 3

Rewrite $\log (x) = 15$ 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 $b^{y} = x$ .

$10^{15} = x$

Step 4

Solve for $x$ .

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Step 4.1

Rewrite the equation as $x = 10^{15}$ .

$x = 10^{15}$

Step 4.2

Raise $10$ to the power of $15$ .

$x = 1000000000000000$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = 1000000000000000$

Decimal Form:

$x = 1 \cdot 10^{15}$