6.1.2

Logarithms

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Logarithms

In order to solve equations with exponential functions in them, we need to define the inverse of the exponential operation.

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Logarithm

  • The logarithmic function is written as logb(x)=y\log_b(x)=y, which is equivalent to x=byx=b^y.
    • bb is called the base of the logarithm.
  • To remember this, think "left to the right equals middle".
    • The base 2 logarithm of the number 8 is 3, so log2(8)=323=8\log_2(8)=3\Leftrightarrow2^3=8
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Range and domain

  • The range of the exponential function is the set of positive real numbers.
  • This means the domain of a logarithm is the set of positive real numbers.
    • So we can only define logb\log_b for x>0x>0.
  • We also specified that the allowed range of values of the base bb of the exponential function was the positive real numbers excluding the number 1.
    • So we can only define logbx\log_bx for b>0,b1b>0,b\ne1.
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Properties

  • For single variables and numbers, we often drop the parenthesis to make logarithms easy to write.
  • The logarithm of the number one is equal to zero for any base, so:
    • logb1=0\log_b1=0
  • For logarithms of base 10, we often drop the base number and write:
    • log10x=logx\log_{10}x=\log x
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Graph

  • The graph of y=logbxy=\log_bx does not cross the yy-axis.
  • The xx--intercept is 1.
  • The orientation of the graph is different for values of b>0b>0 and 0<b<10<b<1 as shown.

Jump to other topics

1Proof

2Algebra & Functions

2.1Powers & Roots

2.2Quadratic Equations

2.3Inequalities

2.4Polynomials

2.5Graphs

2.6Functions

2.7Transformation of Graphs

2.8Partial Fractions (A2 Only)

3Coordinate Geometry

4Sequences & Series

5Trigonometry

6Exponentials & Logarithms

7Differentiation

8Integration

9Numerical Methods

10Vectors

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