INTRODUCTION TO LOGARITHMS

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WHAT YOU SHOULD LEARN:

I can convert logarithmic expressions to exponential expressions and vice

versa.

I can evaluate logarithmic functions.

2.4 I can convert logarithmic expressions to exponential expressions and vice versa.

WHAT IS A LOGARITHM? Definition of a Logarithm

If b > 0, b ≠ 1, and x > 0, then Logarithmic Form Exponential Form

log b x = y iff by = x

base exponent baseexponent

Remember: A logarithm is an exponent!

CONVERT EACH LOGARITHMIC EXPRESSION TO AN EQUIVALENT EXPONENTIAL EXPRESSION.

① Log381 = 4

② Log2 = -3

③ Log10100 = 2

④ Log5 = -3

⑤ Log273 =

34 = 815-3 = 1/125

2-3 = 1/8271/3 = 3

102 = 100

CONVERT EACH EXPONENTIAL EXPRESSION TO AN EQUIVALENT LOGARITHMIC EXPRESSION.

① 92 = 81

② 54 = 625

③ 12-2 =

④ 102 =100

⑤

Log981 = 2log10100 = 2

Log5625 = 4log5√5 = 1/2

log121/144 = -2

THE COMMON LOGARITHM

A logarithm with base 10 or log10 is called a common logarithm.

The common logarithm is often written without the base.

y = logx iff 10y = x

THE NATURAL LOGARITHMRECALL: E ≈ 2.71828…

A logarithm with base e or loge is called a natural logarithm.

The natural logarithm is often written without the base.

y = lnxiff ey = x

CONVERT EACH EXPONENTIAL EXPRESSION TO AN EQUIVALENT LOGARITHMIC EXPRESSION OR VICE VERSA.

① Log x = 2

② Ln 20.0855… ≈ 3

③ e4 ≈ x

④ 106 = 1,000,000

102 = x

e3 ≈ 20.0855…

Ln x = 4

Log 1,000,000 = 6

EVALUATE THE FOLLOWING:① log = x

② log / / = x

③ Log x = 3

④ log x = 0

⑤ log x = -2

⑥ Log x / /

19x = 361x = 2

30 = xx = 1

(1/7)x = 1/49x = 2

9-2 = xx = 1/81

x3 = 216x = 6

X-1/4 = 1/2x = 16

2.5 I can evaluate logarithmic functions in a real world

scenario.

A solution’s pH is given by the function p(t) log(t), where t is the hydronium ion concentration, in moles per liter. A sample of coffee has a pH of 5.0. What is the approximate hydronium ion concentration of the sample?

The wind speed s (in miles per hour) near the center of a tornado is related to the distance d (in miles) the tornado travels by the equation s = 93logd + 65.

a. On March 18, 1925, a tornado whose wind speed was about 280 miles per hour struckthe Midwest. How far did the tornado travel?