Precalculus Examples

$\log_{b} (2) = \frac{1}{6}$

Step 1

Rewrite $\log_{b} (2) = \frac{1}{6}$ in exponential form using the definition of a logarithm. If $x$ and $b$ are positive real numbers and $b \neq 1$ , then $\log_{b} (x) = y$ is equivalent to $b^{y} = x$ .

$b^{\frac{1}{6}} = 2$

Step 2

Solve for $b$ .

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Step 2.1

Raise each side of the equation to the power of $6$ to eliminate the fractional exponent on the left side.

${(b^{\frac{1}{6}})}^{6} = 2^{6}$

Step 2.2

Simplify the exponent.

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Step 2.2.1

Simplify the left side.

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Step 2.2.1.1

Simplify ${(b^{\frac{1}{6}})}^{6}$ .

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Step 2.2.1.1.1

Multiply the exponents in ${(b^{\frac{1}{6}})}^{6}$ .

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Step 2.2.1.1.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$b^{\frac{1}{6} \cdot 6} = 2^{6}$

Step 2.2.1.1.1.2

Cancel the common factor of $6$ .

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Step 2.2.1.1.1.2.1

Cancel the common factor.

$b^{\frac{1}{6} \cdot 6} = 2^{6}$

Step 2.2.1.1.1.2.2

Rewrite the expression.

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

Step 2.2.1.1.2

Simplify.

$b = 2^{6}$

Step 2.2.2

Simplify the right side.

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Step 2.2.2.1

Raise $2$ to the power of $6$ .

$b = 64$