# Logarithmic Functions Examples

### Example 1

Solve the exponential equation 3^{x} – 5 = 15y + 5 for *x* | |

First, isolate the term with *x* in it: 3^{x} = 15*y* + 10 Then take the base-3 logarithm of both sides: log_{3}(3^{x}) = log_{3}(15*y* + 10) Simplify, using the rules of logarithms: *x* = log_{3}(15*y* + 10) *x* = log_{3}(15*y*+10)
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### Example 2

Change the base of log_{4}20 to base 10 and solve to 4 decimal places | |

The change of base formula is log_{a}*x* = log_{b}*x* / log_{b}*a*:
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### Example 3

Find the inverse of the function: y = (7^{x} + 5) ^{1/5} | |

Take the base-*y* log of both sides: log_{y}*y* = log_{y}(7^{x} + 5) ^{1/5} The left side cancels: 1 = log_{y}(7^{x} + 5) ^{1/5} Pull out the exponent using the exponent in log rule: Multiply both sides by 5: 5 = log_{y}(7^{x} + 5) Exponentiate both sides with a base of *y*: *y*^{5} = 7^{x} + 5 Isolate the term with *x*: *y*^{5} – 5 = 7^{x} Take the base-7 log: log_{7}(*y*^{5} – 5) = log_{7}7^{x} The right side cancels: log_{7}(y^{5} – 5) = *x* Switch *x* and *y*: *y* = log_{7}(*x*^{5} – 5) *y* = log_{7}(*x*^{5} – 5)
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### Example 4

Show to 3 decimal places that log_{4} 64 = log_{7} 343 using base-10 logarithms | |

First, use the change of base formula on both sides: Solve each individually: Simplify: 3 = 3 True! | |

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