5.6 and 5.7 logs day 1 and 2.notebook
Logs Day 2
Evaluate:
log5 625 = x
log61 = x
log 1000 = x
log322 = x
May 02, 2016
5.6 and 5.7 logs day 1 and 2.notebook
May 02, 2016
Investigation:
Step 1: Enter the equation y1 = 10x in your calculator. Make a table
of values for y1.
Step 2: Enter the equation y2 = log (10x) and compare the table
values for y1 and y2. What observations can you make? Try
starting your table at different values (including negative values)
and using different decimal increment values.
Step 3: Based on your observations in step 2, what are the
values of the following expressions?
a. log (102.5)
b. log (10-3.2)
c. log (100)
d. log (10x)
5.6 and 5.7 logs day 1 and 2.notebook
May 02, 2016
Step 4: Complete the following statements...
a. If 100 = 102, then log 100 = _____
b. If 400 ≈102.6021, then log ____ = ____
c. If ____ ≈10___, then log 500 ≈_____
d. If y = 10x, then log ____ = ____
Step 5: Use logarithms to solve each equation for x. Check your
answers.
a. 300 = 10x
b. 47 = 10x
c. 0.01 = 10x
d. y = 10x
Step 6: Use a friendly window with a factor of 1 to investigate
the graph of y = log x. Is y = log x a function? What are the
domain and range of y = log x?
5.6 and 5.7 logs day 1 and 2.notebook
Evaluate the following logarithms:
1. Evaluating logs using the calculator
Before evaluating the next one, first estimate what integers it
will be between.
May 02, 2016
5.6 and 5.7 logs day 1 and 2.notebook
Find the inverse of But how do
you solve for y!
Step 7:
Graph y = 10x and draw its inverse on the same set of axes.
Now graph y=log x. What observations can you make?
Step 8:
If f(x) = 10x, then what is f-1(x)?
What is f(f-1(x))?
May 02, 2016
5.6 and 5.7 logs day 1 and 2.notebook
The inverse of logs
Find the inverse of Now graph both f(x) and f ­1(x).
Quickly find the inverse.
May 02, 2016
5.6 and 5.7 logs day 1 and 2.notebook
May 02, 2016
The expression log x is another way
of expressing x as an exponent on the
base 10.
Ten is the common base for logarithms, so
log x is called a "common
logarithm" and is shorthand for writing
log10 x.
You read this as “the logarithm base 10 of x.”
Log x is the exponent you put on 10 to get x.
5.6 and 5.7 logs day 1 and 2.notebook
May 02, 2016
Solve:
4 10x=4650
4x=128
You know that 43 = 64 and 44 = 256, so if 4x = 128,
x must be between 3 and 4. You can rewrite the
equation as x = log4128 by the definition of a logarithm.
But the calculator doesn’t have a built-in logarithm base 4
function. Have we hit a dead end?
5.6 and 5.7 logs day 1 and 2.notebook
Recall that x = log4128 and that 2.1072 was
an approximation for log 128 and 0.6021 was
an approximation for log 4. The numerator
and denominator of the last step above suggest
a more direct way to solve x = log4128.
May 02, 2016
5.6 and 5.7 logs day 1 and 2.notebook
This relationship works because you can
write any number using the inverse
functions of logarithms and exponents.
Doing a composition of functions that
are inverses of each other produces an
output value that is the same as the
input. By definition, the equation 10x = 4
is equivalent to x = log 4.
Substitution from the second equation
into the first equation gives you
10log 4 = 4. More generally, 10log x = x,
which means y = 10x and y = log x are
inverses.
This relationship allows you to rewrite
any logarithmic expression with base 10.
You could also choose to rewrite a
logarithmic expression with
any other base, so
May 02, 2016
5.6 and 5.7 logs day 1 and 2.notebook
Change of Base formula:
Examples:
Homework:
Logs Worksheet II
And Activity 21 WS
May 02, 2016