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Evaluating logarithms

A logarithm is an exponentif there is no base

listed, the common log base is

10. Evaluate, Using a Calculator

Evaluating logarithms

A logarithm is an exponentif there is no base

listed, the common log base is 10. Evaluate

Using the calculator-evaluate to the nearest

hundredth

Using the calculator

The natural logarithm ln

The natural log is base e

The natural logarithm ln

The natural log is base e

Logarithmic exponential

Log base value exponent Log 2 8 3

23 8 Solve for x Log 6 x 3

Logarithmic exponential

Log base value exponent Log 2 8 3

23 8 Solve for x Log 6 x 3 63

x x 216

Change of base law

When not a common logarithm use the change of

base Law

Change of base law

When not a common logarithm use the change of

base Law

Numerical example-evaluate

Laws of Logarithms

Example from previous slide..

But this is also true

Example

Examples expand using log laws

Examples

Single logarithms

Single logarithmsdivision, then multiplication

Express as a single logarithm

Express as a single logarithm

Expansion with substitution

Given

Find

First expand then substitute

Expansion with substitution

First expand then substitute

With numbers.

Rewrite the numbers in terms of factors of 4 and

5 only

With numbers.

Rewrite the numbers in terms of factors of 4 and

5 And use and find

solution

Try page 335 46, 50,52

Solving log equations the domain of y log x

is that xgt0!

Do Now1. solve the exponential equation

2. Solve the logarithmic equation

Solving log equations when the base is a variable

Recall to raise both change to an sides

to the reciprocal power exponential equation

Solving log equations.(combine the last 2

concepts)

- Change to an exponential equation
- Raise to the reciprocal power.
- Check all answers!! (Solutions may be extraneous)

Solving log equations.

- Change to an exponential equation
- Raise to the reciprocal power.

More examples in ex. 1, note that the domain is

x-2gt0 soxgt2

Solutions

1.

2.

Express as a single log first!

Express as a single log first!

-2 is extraneous!

When you cant rewrite using the same base, you

can solve by taking a log of both sides

2x 7 log 2x log 7 x log 2 log 7 x

2.807

Example 4x 15

Example 4x 15

log 4x log 15 x log 4 log15 x log 15/log

4 1.95

Using logarithms to solve exponentials

Here we should ln both sides.

Using logarithms to solve exponentials

Solving with logs-isolate first

Solving with logs-isolate first

Isolate the base term first!

102x 4 21

Isolate the base term first!

102x 4 21 102x 17 log 102xlog 17

Isolate the base term first!

102x 4 21 102x 17 log 102xlog 17 2xlog 10

log 17

Use ( )!

Graphs of exponentials

- Growth and decay
- growth decay

Compound Formula

- Interest rate formula

Compound Formula

- How long will it take 200 to become 250 at 5

interest rate, compounded quarterly

Compound Formula

- How long will it take 200 to become 250 at 5

interest rate, compounded quarterly

solution

- Log both sides and round to the nearest year

CONTINUOUS growth Ex population grows

continuously at a rate of 2 in Allentown. If

Allentown has 10,000 people today, how many years

will it take To have about 11,000 to the nearest

tenth of a year?

CONTINUOUS growth

Solving Log Equations

- To solve use the property for logs w/ the same

base - If logbx logby, then x y

log3(5x-1) log3(x7)

- Solve by decompressing

log3(5x-1) log3(x7)

- 5x 1 x 7
- 5x x 8
- 4x 8
- x 2 and check
- log3(52-1) log3(27)
- log39 log39

Example

- Solve

Example

- Solve

log5x log(x1)log100

- Decompress

log5x log(x1)log100

- (5x)(x1) 100 (product property)
- (5x2 5x) 100
- 5x2 5x-100 0
- x2 x - 20 0 (subtract 100 and

divide by 5) - (x5)(x-4) 0 x-5, x4
- 4x is the only solution

another

- Solve

another

- Solve

One More!log2x log2(x-7) 3

- Solve and check

One More!log2x log2(x-7) 3

- log2x(x-7) 3
- log2 (x2- 7x) 3
- 2log2(x -7x) 23
- x2 7x 8
- x2 7x 8 0
- (x-8)(x1)0
- x8 x -1

2

Graphs of exponentials

- Growth and decay
- growth decay

Inverse functions

- Inverse functions are a reflection in yx

Y2x

Ylog2x

Yx

Domain of y2x is all reals Domain of y log2x

is