Precalculus Examples

$\log_{\frac{2}{3}} (\frac{9}{4}) = a$

Step 1

Rewrite the equation as $a = \log_{\frac{2}{3}} (\frac{9}{4})$ .

$a = \log_{\frac{2}{3}} (\frac{9}{4})$

Step 2

Logarithm base $\frac{2}{3}$ of $\frac{9}{4}$ is $- 2$ .

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Step 2.1

Rewrite as an equation.

$a = \log_{\frac{2}{3}} (\frac{9}{4}) = x$

Step 2.2

Rewrite $\log_{\frac{2}{3}} (\frac{9}{4}) = x$ 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 $b^{y} = x$ .

$a = {(\frac{2}{3})}^{x} = \frac{9}{4}$

Step 2.3

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

$a = {(\frac{2}{3})}^{x} = {(\frac{2}{3})}^{- 2}$

Step 2.4

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

$a = x = - 2$

Step 2.5

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

$a = - 2$