Algebra Examples

$\log_{5} (100)$

Step 1

Rewrite $\log_{5} (100)$ 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 $b = 10$ .

$\frac{\log (100)}{\log (5)}$

Step 2

Logarithm base $10$ of $100$ is $2$ .

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Step 2.1

Rewrite as an equation.

$\frac{\log (100) = x}{\log (5)}$

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 $b^{y} = x$ .

$\frac{10^{x} = 100}{\log (5)}$

Step 2.3

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

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

Step 2.4

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

$\frac{x = 2}{\log (5)}$

Step 2.5

The variable $x$ is equal to $2$ .

$\frac{2}{\log (5)}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$\frac{2}{\log (5)}$

Decimal Form:

$2.86135311 \dots$