Algebra Examples

$\log_{4} (64) = 2 b - 1$

Step 1

Rewrite the equation as $2 b - 1 = \log_{4} (64)$ .

$2 b - 1 = \log_{4} (64)$

Step 2

Logarithm base $4$ of $64$ is $3$ .

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Step 2.1

Rewrite as an equation.

$2 b - 1 = \log_{4} (64) = x$

Step 2.2

Rewrite $\log_{4} (64) = x$ 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 $b^{y} = x$ .

$2 b - 1 = 4^{x} = 64$

Step 2.3

Create expressions in the equation that all have equal bases.

$2 b - 1 = {(2^{2})}^{x} = 2^{6}$

Step 2.4

Rewrite ${(2^{2})}^{x}$ as $2^{2 x}$ .

$2 b - 1 = 2^{2 x} = 2^{6}$

Step 2.5

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

$2 b - 1 = 2 x = 6$

Step 2.6

Solve for $x$ .

$2 b - 1 = x = 3$

Step 2.7

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

$2 b - 1 = 3$

Step 3

Move all terms not containing $b$ to the right side of the equation.

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Step 3.1

Add $1$ to both sides of the equation.

$2 b = 3 + 1$

Step 3.2

Add $3$ and $1$ .

$2 b = 4$

Step 4

Divide each term in $2 b = 4$ by $2$ and simplify.

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Step 4.1

Divide each term in $2 b = 4$ by $2$ .

$\frac{2 b}{2} = \frac{4}{2}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $2$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{2 b}{2} = \frac{4}{2}$

Step 4.2.1.2

Divide $b$ by $1$ .

$b = \frac{4}{2}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Divide $4$ by $2$ .

$b = 2$