Algebra Examples

$\log_{\frac{3}{4}} ({(\frac{1024}{243})}^{- 2})$

Step 1

Change the sign of the exponent by rewriting the base as its reciprocal.

$\log_{\frac{3}{4}} ({(\frac{243}{1024})}^{2})$

Step 2

Apply the product rule to $\frac{243}{1024}$ .

$\log_{\frac{3}{4}} (\frac{243^{2}}{1024^{2}})$

Step 3

Raise $243$ to the power of $2$ .

$\log_{\frac{3}{4}} (\frac{59049}{1024^{2}})$

Step 4

Raise $1024$ to the power of $2$ .

$\log_{\frac{3}{4}} (\frac{59049}{1048576})$

Step 5

Logarithm base $\frac{3}{4}$ of $\frac{59049}{1048576}$ is $10$ .

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Step 5.1

Rewrite as an equation.

$\log_{\frac{3}{4}} (\frac{59049}{1048576}) = x$

Step 5.2

Rewrite $\log_{\frac{3}{4}} (\frac{59049}{1048576}) = x$ 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 $b^{y} = x$ .

${(\frac{3}{4})}^{x} = \frac{59049}{1048576}$

Step 5.3

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

${(\frac{3}{4})}^{x} = {(\frac{3}{4})}^{10}$

Step 5.4

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

$x = 10$

Step 5.5

The variable $x$ is equal to $10$ .

$10$