Algebra Examples

$\log_{0.1} (100)$

Step 1

Rewrite

$\log_{0.1} (100)$ using the change of base formula.

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Step 1.1

The change of base rule can be used if

$a$ and

$b$ are greater than

$0$ and not equal to

$1$ , and

$x$ is greater than

$0$ .

$\log_{a} (x) = \frac{\log_{b} (x)}{\log_{b} (a)}$

Step 1.2

Substitute in values for the variables in the change of base formula, using

$b = 10$ .

$\frac{\log (100)}{\log (0.1)}$

Step 2

Logarithm base

$10$ of

$100$ is

$2$ .

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Step 2.1

Rewrite as an equation.

$\frac{\log (100) = x}{\log (0.1)}$

Step 2.2

Rewrite

$\log (100) = x$ 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

$b^{y} = x$ .

$\frac{10^{x} = 100}{\log (0.1)}$

Step 2.3

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

$\frac{10^{x} = 10^{2}}{\log (0.1)}$

Step 2.4

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

$\frac{x = 2}{\log (0.1)}$

Step 2.5

The variable

$x$ is equal to

$2$ .

$\frac{2}{\log (0.1)}$

Step 3

Log base

$10$ of

$0.1$ is approximately

$- 1$ .

$\frac{2}{- 1}$

Step 4

Divide

$2$ by

$- 1$ .

$- 2$

$\log_{0.1} (100)$

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