Precalculus Examples

$\log_{2} (16) = 3 x + 7$

Step 1

Rewrite the equation as $3 x + 7 = \log_{2} (16)$ .

$3 x + 7 = \log_{2} (16)$

Step 2

Logarithm base $2$ of $16$ is $4$ .

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Step 2.1

Rewrite as an equation.

$3 x + 7 = \log_{2} (16) = x$

Step 2.2

Rewrite $\log_{2} (16) = x$ 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 $b^{y} = x$ .

$3 x + 7 = 2^{x} = 16$

Step 2.3

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

$3 x + 7 = 2^{x} = 2^{4}$

Step 2.4

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

$3 x + 7 = x = 4$

Step 2.5

The variable $x$ is equal to $4$ .

$3 x + 7 = 4$

Step 3

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

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Step 3.1

Subtract $7$ from both sides of the equation.

$3 x = 4 - 7$

Step 3.2

Subtract $7$ from $4$ .

$3 x = - 3$

Step 4

Divide each term in $3 x = - 3$ by $3$ and simplify.

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Step 4.1

Divide each term in $3 x = - 3$ by $3$ .

$\frac{3 x}{3} = \frac{- 3}{3}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $3$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{- 3}{3}$

Step 4.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 3}{3}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Divide $- 3$ by $3$ .

$x = - 1$