Basic Technical Mathematics, 11th Edition

13.2 Logarithmic Functions

• Exponential Form • Logarithmic Form • Logarithmic Function • Graphing Logarithmic Functions • Features of Logarithmic Functions • Inverse Functions

For many uses in mathematics and for many applications, it is necessary to express the exponent x in the exponential function $y\hspace{0.17em}=\hspace{0.17em}{b}^x$ in terms of y and the base b. This is done by defining a logarithm. Therefore, if $y\hspace{0.17em}=\hspace{0.17em}{b}^x\hspace{0.17em},\hspace{0.17em}$ the exponent x is the logarithm of the number y to the base b. We write this as

$$\text{If }y\hspace{0.17em}=\hspace{0.17em}{b}^x\hspace{0.17em},\hspace{0.17em}\text{ then }x\hspace{0.17em}=\hspace{0.17em}{\log }_{b}y\hspace{0.17em}.\hspace{0.17em}$$ (13.2)

This means that x is the power to which the base b must be raised in order to equal the number y. That is, x is a logarithm, and a logarithm is an exponent. As with the exponential function, for the equation $x\hspace{0.17em}=\hspace{0.17em}{\text{log}}_{b}y\hspace{0.17em},\hspace{0.17em}x$ may be ...