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# ch 25- logs and base

### comman/naturals logs, change of base. equations and applications

chapter objecives | |
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Identify common logarithms. | Calculate y in y=log4(1/4) Start with y=log4(1/4) Use the Exponential Function on both sides: 4^y=4^( log4(1/4) ) Simplify:4y = 1/4 Now1/4 = 4-1 4y = 4-1 y = -1 |

Approximate common logarithms using a calculator. | Logarithms with base 10 are called common logarithms.When the base is not indicated, base 10 is implied.The log key on the graphing calculator will calculate the common (or base 10) logarithm. 2nd log will calculate the antilogarithm or 10x |

Evaluate common logarithms of powers of 10. | Simplify log(100). Since 100 = 102, then log(100) = log(102) = 2, because "log(100) = y" means "10 y = 100 = 102", so y = 2. log(100) = 2 So I'll plug this into my calculator, use the "LOG" key log(98) = 1.99122 |

Identify natural logarithms. | Logarithms with base e are called natural logarithms. Natural logarithms are denoted by ln. On the graphing calculator, the base e logarithm is the ln key. |

Approximate natural logarithms using a calculator. | Remember that "ln( )" means the base-e log, so "ln(e4.5)" might be thought of as "loge(e4.5)". The Relationship says that "ln(e4.5) = y" means "e y = e4.5", so y = 4.5, and: ln(e4.5) = 4.5 |

Evaluate natural logarithms of powers of e. | ln 23x + 1 = 5. (3x + 1) ln 2 = 5 3x ln 2 + ln 2 = 5 3x ln 2 = 5 − ln 2 x = 5 − ln 2/3 ln |

Approximate logarithms for bases other than 10 or e using the change of base formula. | To enter a logarithm with a different base on the graphing calculator, use the Change of Base Formula: logb(x)=logd(x) / logd(x) |

Solve exponential equations. | a) log ab c = log a + log b − log c b) log ab2 c4 = log a + 2 log b − 4 log c |

Solve logarithmic equations. | Solve for x in the equation Ln(x)=8. Let both sides be exponents of the base e. when the base of the exponent and the base of the logarithm are the same, the left side can be written x x=e^8 |

Solve problems that can be modeled by exponential and logarithmic equations. | calculate the nearest hundredth of a year, how long it takes for an amount of money to double if interest is compounded continuously at 4.3%. You want to know when 2p = pe^0.043t e^0.043t = 2 0.043t = ln2 t = ln2//0.043 which is about 16.1197018 |

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