Logs and Properties of Logs

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LOGARITHMS

The inverse of the exponential function-- Logarithms

Below is the graph of

y = 2^x

and its inverse which we defined as

y = log₂ x

$Graphs of y=2^x and log base 2 of x. They are reflections of each other across the line y = x.$

We see that the logarithm function y = log_bx has the following properties:

The x - intercept is (1,0).
There is a vertical asymptote at x = 0.
The domain is {x | x > 0}.
The range is all real numbers.
The graph goes through (b,1).

Evaluating Logarithms

Example

Evaluate

log₃ 81

Solution:

We run the "hook" as shown below

$Shows log_3 81 = y and a hook starting at 3, going through y, then through =, then to 81.$

and write

3^y = 81

so that

y = 4

Exercises

Evaluate the following

log₁₀ 100,000,000
log₅ (1/125)
log₂₇ 9

Example

Solve

log₉ x = 2

Solution

We write

x = 9² = 81

Exercises

Solve

log₂ x = 5
log_b 64 = 3

Inverse Properties of Logs

Since logs and exponents cancel each other we have:

b^log^b^x= x

and

log_b b^x = x

Example

2^log2³ = 3

and

log₄ 4⁵ = 5

Three Properties of Logs

Property 1: log_b (uv) = log_b u + log_b v (The Product to Sum Rule)

Property 2: log_b (u/v) = log_b u - log_b v (The Quotient to Difference Rule)

Property 3: log_b u^r = rlog_b u (The Power Rule)

Proof of the power rule

We have the rule for exponents:

$ b^{log_b u^r} = u^r = (b^{log_b u})^r = b^{r log_b u} $

Canceling the b we get

log_b u^r = rlog_b u

Example

Expand:

log₂ (xy²/z)

by property 2 we have:

log₂ (xy²) - log₂ z

by property 1 we have

log₂ x + log₂ y² - log₂ z

By property 3 we have

log₂ x + 2 log₂ y - log₂ z

Exercise

Try to expand:

$ log_5 (25 \sqrt{\frac{x}{y^3}}) $

Example

Write as a single logarithm:

4 log₂ x - 1/2 log₂ y + log₂ z

Solution:

We first use property 3 to write:

log₂ x⁴ - log₂ y^1/2 + log₂ z

Now we use property 2:

log₂ x⁴/y^1/2 + log₂ z

Finally, we use property 3:

                   x⁴z
        log₂ (           )
                  y^1/2

Exercise

Write the following as a single logarithm:

1/3 log₃ x + 2 log₃ y - 3 log₃ z

Example

Suppose that

log₂ 3 = 1.58

and that

log₂ 5 = 2.32

Find

log₂ 90

Solution

Since

90 = (2)(5)(3²)

We have

log₂ 90 = log₂ (2)(5)(3²) = log₂ 2 + log₂ 5 + log₂ 3²

= 1 + 2.32 + 2log₂ 3

= 1 + 2.32 + 2(1.58) = 6.48

Exercise

Find

log₂ 40/27

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