Evaluating logarithms A logarithm is an exponent…if there is no base listed, the common log base is 10. Evaluate, Using a Calculator:

Evaluating logarithms A logarithm is an exponent…if 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

Change of base law: When not a common logarithm use the change of base

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 logarithms division, 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 x>0! Do Now:1. 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-2>0 so…x>2

Solutions: 1. 2.

Express as a single log first!

Express as a single log first! -2 is extraneous!

When you can’t 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 102x=log 17

Isolate the base term first! 102x +4 = 21 102x = 17 log 102x=log 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:

If logbx = logby, then x = y Solving Log Equations To solve use the property for logs w/ the same base: If logbx = logby, then x = y

Solve by decompressing log3(5x-1) = log3(x+7) Solve by decompressing

5x – 1 = x + 7 5x = x + 8 4x = 8 x = 2 and check log3(5x-1) = log3(x+7) 5x – 1 = x + 7 5x = x + 8 4x = 8 x = 2 and check log3(5*2-1) = log3(2+7) log39 = log39

Example: Solve:

Example: Solve:

log5x + log(x+1)=log100 Decompress

log5x + log(x+1)=log100 x2 + x - 20 = 0 (subtract 100 and divide by 5) (5x)(x+1) = 100 (product property) (5x2 + 5x) = 100 5x2 + 5x-100 = 0 x2 + x - 20 = 0 (subtract 100 and divide by 5) (x+5)(x-4) = 0 x=-5, x=4 4=x 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)(x+1)=0 x=8 x= -1 2

Graphs of exponentials Growth and decay: growth decay

Inverse functions Inverse functions are a reflection in y=x Y=log2x Y=x Domain of y=2x is all reals Domain of y = log2x is