Finite Math Examples

$\log_{b} (3) = 0.517$

Step 1

Rewrite $\log_{b} (3) = 0.517$ 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 $b^{y} = x$ .

$b^{0.517} = 3$

Step 2

Solve for $b$ .

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Step 2.1

Convert the decimal exponent to a fractional exponent.

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Step 2.1.1

Convert the decimal number to a fraction by placing the decimal number over a power of ten. Since there are $3$ numbers to the right of the decimal point, place the decimal number over $10^{3}$ $(1000)$ . Next, add the whole number to the left of the decimal.

$b^{0 \frac{517}{1000}} = 3$

Step 2.1.2

Convert $0 \frac{517}{1000}$ to an improper fraction.

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Step 2.1.2.1

A mixed number is an addition of its whole and fractional parts.

$b^{0 + \frac{517}{1000}} = 3$

Step 2.1.2.2

Add $0$ and $\frac{517}{1000}$ .

$b^{\frac{517}{1000}} = 3$

Step 2.2

Raise each side of the equation to the power of $\frac{1}{0.517}$ to eliminate the fractional exponent on the left side.

${(b^{\frac{517}{1000}})}^{\frac{1}{0.517}} = 3^{\frac{1}{0.517}}$

Step 2.3

Simplify the exponent.

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Step 2.3.1

Simplify the left side.

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Step 2.3.1.1

Simplify ${(b^{\frac{517}{1000}})}^{\frac{1}{0.517}}$ .

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Step 2.3.1.1.1

Multiply the exponents in ${(b^{\frac{517}{1000}})}^{\frac{1}{0.517}}$ .

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Step 2.3.1.1.1.1

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

$b^{\frac{517}{1000} \cdot \frac{1}{0.517}} = 3^{\frac{1}{0.517}}$

Step 2.3.1.1.1.2

Cancel the common factor of $0.517$ .

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Step 2.3.1.1.1.2.1

Factor $0.517$ out of $517$ .

$b^{\frac{0.517 (1000)}{1000} \cdot \frac{1}{0.517}} = 3^{\frac{1}{0.517}}$

Step 2.3.1.1.1.2.2

Cancel the common factor.

$b^{\frac{0.517 \cdot 1000}{1000} \cdot \frac{1}{0.517}} = 3^{\frac{1}{0.517}}$

Step 2.3.1.1.1.2.3

Rewrite the expression.

$b^{\frac{1000}{1000}} = 3^{\frac{1}{0.517}}$

Step 2.3.1.1.1.3

Divide $1000$ by $1000$ .

$b^{1} = 3^{\frac{1}{0.517}}$

Step 2.3.1.1.2

Simplify.

$b = 3^{\frac{1}{0.517}}$

Step 2.3.2

Simplify the right side.

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Step 2.3.2.1

Simplify $3^{\frac{1}{0.517}}$ .

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Step 2.3.2.1.1

Divide $1$ by $0.517$ .

$b = 3^{1.93423597}$

Step 2.3.2.1.2

Raise $3$ to the power of $1.93423597$ .

$b = 8.37269162$