6.1.3

Properties of Logarithms

Test yourself

Properties of Logarithms

We can use the properties of exponents to find ways of combining logarithms of the same base.

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Inverse

  • The definition of a logarithm as the inverse of an exponent allows us to write the following expressions:
    • logbbx=x\log_bb^x = x
    • blogbx=x,x>0b^{\log_bx} = x, x>0
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Product rule

  • Adding logarithms of the same base follows the rule:
    • logbx+logby=logbxy\log_bx+\log_by=\log_bxy
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Quotient rule

  • Subtracting logarithms of the same base follows the rule:
    • logbxlogby=logbxy\log_bx-\log_by=\log_b\frac{x}{y}
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Power rule

  • The logarithm of a number raised to a power can be rewritten as follows:
    • logbxy=ylogbx\log_bx^y=y\log_bx

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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