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8.4 Logarithms p. 486

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Daily Question: How do you evaluate a logarithm?

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**Logarithmic Function x = log b y read: “x equals log base b of y”**

The inverse of an exponential function is a logarithmic function. x = log b y read: “x equals log base b of y”

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**These two equations are equivalent**

We can convert exponential equations to logarithmic equations and vice versa, using this: y = bx x = logby These two equations are equivalent

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y = bx x = logby Exponential form logarithmic form

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**If y = bx then x = logby Another way to “read” logs:**

unknown If y = bx then x = logby base base Another way to “read” logs: “What is the exponent of b that gives you y?” where y > 0 b > 0 b 1 technical stuff

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**Convert to exponential form**

“What is the exponent of 3 that gives you 5?” 1) 2) 3)

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**Convert to logarithmic form**

4) 5) 6)

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**Now that we can convert between the two forms we can simplify logarithmic expressions.**

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**Simplify “What is the exponent of that gives you 32?”**

7) log2 32 2? = 32 ? = 5 “What is the exponent of 3 that gives you 27?” 8) log3 27 3? = 27 ? = 3 9) log4 2 4? = 2 ? = 0.5 10) log3 1 3? = 1 ? = 0

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Evaluate try:

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**Common Logarithm A common logarithm is a logarithm that is base 10.**

When a logarithm is base 10, we don’t write the base. log10 = log We like base 10 because we can evaluate it in our calculator. (Use the LOG button)

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**Common logs and natural logs with a calculator**

log10 button ln button

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**Evaluate with a calculator**

= 1 11) log10 10 = 12) 2 log10 2.5 no solution 13) log10 (-2) Remember this means 10? = -2

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**Try these using your calculators:**

10x = 85 10x = 1.498 10x = -5.5

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**Natural Exponential Function**

y = ex Natural Base ln e = 1

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**y = ex x = loge y x = ln y Natural Logarithmic Function**

the natural logarithmic function is the inverse of the natural exponential function

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**Convert to natural logarithmic form:**

a. 10 = ex b. 14 = e 2x Convert to natural exponential form: a. ln 4 = 1.386… b. ln 6 = 1.792…

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Evaluate: ln x a. x = b. x = ½ c. x = -1 .693 -0.693 undefined

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Solving natural bases 1.) ex+7 = 98 2.) 4e3x-5 = 72 3.) ln x3 - 5 = 1

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page 490 #16-64 x4 Homework

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