Precalculus Examples

$6 \log ({(a)}^{t}) - \log ({(a)}^{s})$

Step 1

Simplify each term.

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Step 1.1

Simplify $6 \log (a^{t})$ by moving $6$ inside the logarithm.

$\log ({(a^{t})}^{6}) - \log ({(a)}^{s})$

Step 1.2

Multiply the exponents in ${(a^{t})}^{6}$ .

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Step 1.2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\log (a^{t \cdot 6}) - \log ({(a)}^{s})$

Step 1.2.2

Move $6$ to the left of $t$ .

$\log (a^{6 t}) - \log ({(a)}^{s})$

$\log (a^{6 t}) - \log (a^{s})$

Step 2

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

$\log (\frac{a^{6 t}}{a^{s}})$

Step 3

Cancel the common factor of $a^{6 t}$ and $a^{s}$ .

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Step 3.1

Factor $a^{s}$ out of $a^{6 t}$ .

$\log (\frac{a^{s} a^{6 t - s}}{a^{s}})$

Step 3.2

Cancel the common factors.

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Step 3.2.1

Multiply by $1$ .

$\log (\frac{a^{s} a^{6 t - s}}{a^{s} \cdot 1})$

Step 3.2.2

Cancel the common factor.

$\log (\frac{a^{s} a^{6 t - s}}{a^{s} \cdot 1})$

Step 3.2.3

Rewrite the expression.

$\log (\frac{a^{6 t - s}}{1})$

Step 3.2.4

Divide $a^{6 t - s}$ by $1$ .

$\log (a^{6 t - s})$