Precalculus Examples

$\log_{3} (2 x - 1) - 3$

Step 1

Rewrite $\log_{3} (2 x - 1)$ 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 $0$ .

$\log_{a} (x) = \frac{\log_{b} (x)}{\log_{b} (a)} - 3$

Step 1.2

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

$\frac{\log (2 x - 1)}{\log (3)} - 3$

Step 2

To write $- 3$ as a fraction with a common denominator, multiply by $\frac{\log (3)}{\log (3)}$ .

$\frac{\log (2 x - 1)}{\log (3)} - 3 \cdot \frac{\log (3)}{\log (3)}$

Step 3

Simplify terms.

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Step 3.1

Combine $- 3$ and $\frac{\log (3)}{\log (3)}$ .

$\frac{\log (2 x - 1)}{\log (3)} + \frac{- 3 \log (3)}{\log (3)}$

Step 3.2

Combine the numerators over the common denominator.

$\frac{\log (2 x - 1) - 3 \log (3)}{\log (3)}$

Step 4

Simplify the numerator.

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Step 4.1

Simplify $- 3 \log (3)$ by moving $3$ inside the logarithm.

$\frac{\log (2 x - 1) - \log (3^{3})}{\log (3)}$

Step 4.2

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

$\frac{\log (2 x - 1) - \log (27)}{\log (3)}$

Step 4.3

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

$\frac{\log (\frac{2 x - 1}{27})}{\log (3)}$