Change of Base Formula with Solved Examples

The Change of base formula helps to rewrite the logarithm in terms of another base log. Change of base formula is used in the evaluation of log and have another base than 10.

$\log_{b} x = \frac{\log_{d} x}{\log_{d} b}$

Solved Examples

Example 1:

Solve: 2

\begin{array}{l} \log_{4} 29 \end{array}

Solution:

Given,

\begin{array}{l} \log_{4} 29 \end{array}

Using logarithm change of base formula,

\begin{array}{l} l o g_{b} x \end{array}

\begin{array}{l} \frac{l o g_{d} x}{l o g_{d} b} \end{array}

\begin{array}{l} \log_{4} 29 \end{array}

= 2

\begin{array}{l} \times \end{array}

\begin{array}{l} \frac{l o g_{10} 29}{l o g_{10} 4} \end{array}

\begin{array}{l} \log_{4} 29 \end{array}

= 2 x 2.43 = 4.86

Example 2: Simply:

\begin{array}{l} \log_{32} 16 \end{array}

Solution:

Given,

\begin{array}{l} \log_{32} 16 \end{array}

Using change of base formula,

\begin{array}{l} \log_{32} 16 = \frac{\log_{10} 16}{\log_{10} 32} \\ = \frac{\log_{10} 2^{4}}{\log_{10} 2^{5}} \\ = \frac{4 \log_{10} 2}{5 \log_{10} 2} \\ = \frac{4}{5} \end{array}

FORMULAS Related Links
Triangle Formula Sides	Tangential Velocity Formula
Range Formula	Central Angle
Perimeter Of Trapezoid Formula	Relative Standard Deviation
Formula For Perimeter Of A Rectangle	Perpendicular Formula
Gregory Newton Formula	Summation Formulas

FORMULAS Related Links

Triangle Formula Sides

Tangential Velocity Formula

Range Formula

Central Angle

Perimeter Of Trapezoid Formula

Relative Standard Deviation

Formula For Perimeter Of A Rectangle

Perpendicular Formula

Gregory Newton Formula

Summation Formulas

Change of Base Formula

Solved Examples

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