8.4 Logarithms p. 486

Daily Question: How do you evaluate a logarithm?

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”

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

y = bx x = logby Exponential form logarithmic form

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

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

Convert to logarithmic form 4) 5) 6)

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

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

Evaluate try:

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)

Common logs and natural logs with a calculator log10 button ln button

Evaluate with a calculator = 1 11) log10 10 = 0.7959 12) 2 log10 2.5 no solution 13) log10 (-2) Remember this means 10? = -2

Try these using your calculators: 10x = 85 10x = 1.498 10x = -5.5

Natural Exponential Function y = ex Natural Base ln e = 1

y = ex x = loge y x = ln y Natural Logarithmic Function the natural logarithmic function is the inverse of the natural exponential function

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…

Evaluate: ln x a. x = 2 b. x = ½ c. x = -1 .693 -0.693 undefined

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

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