5.2 Logarithmic Functions

5.2 Logarithmic Functions

Definition of Logarithm

  • A logarithm is defined as:

    • y=extlogbxy = ext{log}_b x

    • This holds true if and only if:

      • b > 0

      • b<br>eq1b <br>eq 1

      • x > 0

  • Note: extlogbxext{log}_b x is defined only for positive values of xx.

Examples

  • Example 1: Express each equation in logarithmic form.

    • (a) 35=24335 = 243

    • extlog3243=5ext{log}_{3} 243 = 5

    • Explanation: This means that 3 raised to the power of 5 equals 243.

    • (b) 5−3=rac11255^{-3} = rac{1}{125}

    • extlog5rac1125=−3ext{log}_{5} rac{1}{125} = -3

    • Explanation: Indicates that 5 raised to the power of -3 equals 1/125.

Logarithmic Notation

  • extlogx=extlog10xext{log} x = ext{log}_{10} x

  • Common logarithm

  • extlnx=extlogexext{ln} x = ext{log}_{e} x

Laws of Logarithms

  • If xx and yy are positive numbers, then:

    1. extln(xy)=extlnx+extlnyext{ln}(xy) = ext{ln} x + ext{ln} y

    2. extln(racxy)=extlnx−extlnyext{ln}\bigg( rac{x}{y}\bigg) = ext{ln} x - ext{ln} y

    3. extln(xb)=bextlnxext{ln}(x^b) = b ext{ln} x

    4. extln(1)=0ext{ln}(1) = 0

    5. extln(e)=1ext{ln}(e) = 1

  • Warning:

    • extln(racxy)<br>eqextlnx+extlnyext{ln}\bigg( rac{x}{y}\bigg) <br>eq ext{ln} x + ext{ln} y

    • extln(xb)<br>eqextlnxbext{ln}(x^b) <br>eq ext{ln} x^b

Properties of Logarithmic Functions

  • The function defined by f(x)=extlogbxf(x) = ext{log}_b x (where b > 0 and beq1b eq 1) has the following properties:

    1. Domain:

    • extDomain=(0,extinfinity)ext{Domain} = (0, ext{infinity})

    1. Range:

    • extRange=(−extinfinity,extinfinity)ext{Range} = (- ext{infinity}, ext{infinity})

    1. Graph Behavior:

    • The graph passes through the point (1, 0) since extlogb1=0ext{log}_b 1 = 0

    1. Continuity:

    • The function is continuous on (0,extinfinity)(0, ext{infinity}).

    1. Monotonicity:

    • It is increasing if b > 1

    • It is decreasing if 0 < b < 1

Inverse