Precalculus Examples

$\log_{5} (x - 3) + \log_{5} (\log_{5} (10))$

Step 1

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

$\log_{5} ((x - 3) \log_{5} (10))$

Step 2

Apply the distributive property.

$\log_{5} (x \log_{5} (10) - 3 \log_{5} (10))$

Step 3

Simplify $- 3 \log_{5} (10)$ by moving $3$ inside the logarithm.

$\log_{5} (x \log_{5} (10) - \log_{5} (10^{3}))$

Step 4

Raise $10$ to the power of $3$ .

$\log_{5} (x \log_{5} (10) - \log_{5} (1000))$

Step 5

Rewrite $\log_{5} (x \log_{5} (10) - \log_{5} (1000))$ using the change of base formula.

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Step 5.1

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 5.2

Substitute in values for the variables in the change of base formula, using $b = 10$ .

$\frac{\log (x \log_{5} (10) - \log_{5} (1000))}{\log (5)}$

Step 6

Simplify the numerator.

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Step 6.1

Simplify each term.

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Step 6.1.1

Rewrite $\log_{5} (10)$ using the change of base formula.

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Step 6.1.1.1

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 $0$ .

$\frac{\log (x \log_{a} (x) = \frac{\log_{b} (x)}{\log_{b} (a)} - \log_{5} (1000))}{\log (5)}$

Step 6.1.1.2

Substitute in values for the variables in the change of base formula, using $b = 10$ .

$\frac{\log (x \frac{\log (10)}{\log (5)} - \log_{5} (1000))}{\log (5)}$

Step 6.1.2

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

$\frac{\log (x \frac{1}{\log (5)} - \log_{5} (1000))}{\log (5)}$

Step 6.1.3

Combine $x$ and $\frac{1}{\log (5)}$ .

$\frac{\log (\frac{x}{\log (5)} - \log_{5} (1000))}{\log (5)}$

Step 6.1.4

Rewrite $\log_{5} (1000)$ using the change of base formula.

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Step 6.1.4.1

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 $0$ .

$\frac{\log (\frac{x}{\log (5)} - \log_{a} (x) = \frac{\log_{b} (x)}{\log_{b} (a)})}{\log (5)}$

Step 6.1.4.2

Substitute in values for the variables in the change of base formula, using $b = 10$ .

$\frac{\log (\frac{x}{\log (5)} - \frac{\log (1000)}{\log (5)})}{\log (5)}$

Step 6.1.5

Logarithm base $10$ of $1000$ is $3$ .

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Step 6.1.5.1

Rewrite as an equation.

$\frac{\log (\frac{x}{\log (5)} - \frac{\log (1000) = x}{\log (5)})}{\log (5)}$

Step 6.1.5.2

Rewrite $\log (1000) = x$ 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 $b^{y} = x$ .

$\frac{\log (\frac{x}{\log (5)} - \frac{10^{x} = 1000}{\log (5)})}{\log (5)}$

Step 6.1.5.3

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

$\frac{\log (\frac{x}{\log (5)} - \frac{10^{x} = 10^{3}}{\log (5)})}{\log (5)}$

Step 6.1.5.4

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

$\frac{\log (\frac{x}{\log (5)} - \frac{x = 3}{\log (5)})}{\log (5)}$

Step 6.1.5.5

The variable $x$ is equal to $3$ .

$\frac{\log (\frac{x}{\log (5)} - \frac{3}{\log (5)})}{\log (5)}$

Step 6.2

Combine the numerators over the common denominator.

$\frac{\log (\frac{x - 3}{\log (5)})}{\log (5)}$