1. Warm-Up Algebra 2 Honors 2/12/18
Tell whether the exponential functions below show growth or decay and name the growth factor.
f(x)=5(2)x f(x)= 290(.765)x
f(x)= 6(1-.025)x f(x)=44(.4)x
3. f(x)= .2(1+.6)x f(x)= 4x

2. Try this!
How many times would you have to double $1 before you had $8?
1(2x) = 8.
You know 23 = 8. So you would have to double the dollar 3 times to have $8.
How many times would it take to get to 512?
What about 16,384?

3. Notes: Logarithms
Just as we can undo addition with subtraction, multiplication with division, powers with roots, we need an inverse operation that undoes raising a base to an exponent equation such as the ones we just worked with!
This operation is called finding the logarithm.
A logarithm is the exponent to which a specified base is being raised to obtain a given value.

4. We can interchange between logs and exponential functions like this:
Read logb a= x, as “the log base b of a is x.” Notice that the log is the exponent.
Reading Math

5. Example 1: Converting from Exponential to Logarithmic Form
Write each exponential equation in logarithmic form.
Exponential Equation
Logarithmic Form
35 = 243
25 = 5
104 = 10,000
6–1 =
ab = c
The base of the exponent becomes the base of the logarithm.
log3243 = 5 1 2 1 2
log255 =
The exponent is the logarithm.
log1010,000 = 4 1 6
log = –1 1 6
An exponent (or log) can be negative.
logac =b
The log (and the exponent) can be a variable.

6. Write each exponential equation in logarithmic form.
Check It Out! Example 1
Write each exponential equation in logarithmic form.
Exponential Equation
Logarithmic Form
92= 81
33 = 27
x0 = 1(x ≠ 0)
The base of the exponent becomes the base of the logarithm. a.
log981 = 2 b.
log327 = 3
The exponent of the logarithm.
The log (and the exponent) can be a variable. c.
logx1 = 0

7. Example 2: Converting from Logarithmic to Exponential Form
Write each logarithmic form in exponential equation.
Logarithmic Form
Exponential Equation
log99 = 1
log = 9
log82 =
log = –2
logb1 = 0
The base of the logarithm becomes the base of the power.
91 = 9
The logarithm is the exponent.
29 = 512 1 3 1 3
8 = 2
A logarithm can be a negative number. 1 1 16
4–2 = 16
Any nonzero base to the zero power is 1.
b0 = 1

8. Write each logarithmic form in exponential equation.
Check It Out! Example 2
Write each logarithmic form in exponential equation.
Logarithmic Form
Exponential Equation
log1010 = 1
log12144 = 2
log 8 = –3
The base of the logarithm becomes the base of the power.
101 = 10
122 = 144
The logarithm is the exponent. 1 2 –3
= 8
An logarithm can be negative. 1 2

9. Let’s talk about it

10. A logarithm with base 10 is called a common logarithm
A logarithm with base 10 is called a common logarithm. If no base is written for a logarithm, the base is assumed to be 10.
For example, log 5 = log105.
This is similar to not having to write x1, 1x, 2 𝑥 , x+0, 𝑥 if it isn’t written, it’s implied!

11. Example 3A: Evaluating Logarithms by Using Mental Math
Evaluate by using mental math.
log 0.01
10? = 0.01
The log is the exponent.
10–2 = 0.01
Think: What power of 10 is 0.01?
log 0.01 = –2

12. Example 3B: Evaluating Logarithms by Using Mental Math
Evaluate by using mental math.
log5 125
5? = 125
The log is the exponent.
53 = 125
Think: What power of 5 is 125?
log5125 = 3