1. Day 4: more solving equations
Logarithms
Day 4: more solving equations

2. Notes: Natural Log
What is the approximate value of 𝜋? What is the approximate value of e? In math, e is known as Euler’s number. It has an approximate value of 𝑙𝑜𝑔 𝑒 is known as the “natural log” which is represented 𝑙𝑛
This definition means that e is the unique number with the property that the area of the region bounded by the hyperbola , the x-axis, and the vertical lines  and  is 1. In other words,
(3)
The notation  is used in physics and engineering to denote the natural logarithm, while mathematicians commonly use the notation . In this work,  denotes a natural logarithm, whereas  denotes the common logarithm.
There are a number of notational conventions in common use for indication of a power of a natural logarithm. While some authors use  (i.e., using a trigonometric function-like convention), it is also common to write .
Common and natural logarithms can be expressed in terms of each other as
(4)
(5)
The natural logarithm is especially useful in calculus because its derivative is given by the simple equation
(6)
whereas logarithms in other bases have the more complicated derivative

3. Exponential equations
𝑃=𝑃0𝑒 𝑟𝑡 , where 𝑃0= initial value
𝑟= the interest rate as a decimal
𝑡= time in years
P= amount after t years

4. Notes: Natural Log
Examples:
𝑙𝑛10=
ln 𝑥 =4
𝑒 2𝑥−1 =12

5. Savings
You save up some money from your high school job and the day after Graduation you open a mutual fund type savings account with $ The account earns 6%, compounded continously.
write an equation to give you the amount you have in the account after t years.

6. Use you equation to answer the following questions
How much money is in the account after:
5 years?
10 years?
50 years?
Set up and solve an equation to determine when you will have:
$10,000?
$25,000?

7. How do I solve for x? 𝑙𝑜𝑔 4 𝑥+6 + 𝑙𝑜𝑔 4 8 =2
𝑙𝑜𝑔 4 𝑥+6 + 𝑙𝑜𝑔 4 8 =2
Do I have more than 1 logarithm?
Do I have a single logarithm or exponential?
X = -4
Condense
Can I solve for x in the current form? Should I change the logarithm to an exponential or the exponential to a logarithm?
Rewrite and then Solve for x

8. How do I solve for x? 𝑙𝑜𝑔 5 2𝑥 − 𝑙𝑜𝑔 5 2 =2
𝑙𝑜𝑔 5 2𝑥 − 𝑙𝑜𝑔 5 2 =2
Do I have more than 1 logarithm?
Do I have a single logarithm or exponential?
X =
Condense
Can I solve for x in the current form? Should I change the logarithm to an exponential or the exponential to a logarithm?
Rewrite and then Solve for x

9. How do I solve for x? −3∙ 9 𝑥−3 =−75 Do I have more than 1 logarithm?
Do I have a single logarithm or exponential?
X = 4.465
Condense
Can I solve for x in the current form? Should I change the logarithm to an exponential or the exponential to a logarithm?
Rewrite and then Solve for x

10. Notes: Solving for X Ex 2: 𝑙𝑜𝑔 4 𝑥+30 − 𝑙𝑜𝑔 4 𝑥 =3
Step 1: Condense 𝑙𝑜𝑔 4 𝑥+30 𝑥 =3
Step 2: Rewrite 𝑥+30 𝑥 = 4 3
Step 3: Solve x = 0.48

11. Notes: Solving for X Ex. 1: log(x) + log(8) = 3
Step 1: Condense log(8x) = 3
Step 2: Rewrite 8x = 10 3
Step 3: Solve for x x = 125

12. Condense or Expand using log properties 𝒍𝒐𝒈 𝟓 𝒙+𝟏 + 𝒍𝒐𝒈 𝟓 𝒙
I can solve exponential equations using properties of logarithms
Solve for x:
4+𝟑∙ 𝟐 𝒙 =𝟐𝟓
𝒍𝒐𝒈 𝟒 𝟐𝒙+𝟒 +𝟐=𝟒
Condense or Expand using log properties
𝒍𝒐𝒈 𝟓 𝒙+𝟏 + 𝒍𝒐𝒈 𝟓 𝒙
𝒍𝒐𝒈 𝟑 𝟒 𝒙 𝟐 𝒚 𝟑

13. I can solve exponential equations using properties of logarithms
Solve for x:
4+𝟑∙ 𝟐 𝒙 =𝟐𝟓
x = 2.81
𝒍𝒐𝒈 𝟒 𝟐𝒙+𝟒 +𝟐=𝟒
x = 6
Condense or Expand using log properties
𝒍𝒐𝒈 𝟓 𝒙+𝟏 + 𝒍𝒐𝒈 𝟓 𝒙
𝒍𝒐𝒈 𝟓 ( 𝒙 𝟐 +𝒙)
𝒍𝒐𝒈 𝟑 𝟒 𝒙 𝟐 𝒚 𝟑 = 𝒍𝒐𝒈 𝟑 𝟒 +𝟐 𝒍𝒐𝒈 𝟑 𝒙 −𝟑 𝒍𝒐𝒈 𝟑 (𝒚)

14. 𝑥=28 𝑥=9 𝑥=4 𝑥=45.25 𝑥=2 Practice 𝑙𝑜𝑔 5 5𝑥−15 =3 2 𝑙𝑜𝑔 3 𝑥=4
Solve each equation for x. Note: you may need to condense first!
𝑙𝑜𝑔 5 5𝑥−15 =3
2 𝑙𝑜𝑔 3 𝑥=4
𝑙𝑜𝑔 2 4𝑥−12 −3=−1
𝑙𝑜𝑔 8 𝑥+ 𝑙𝑜𝑔 8 2𝑥 =4
𝑙𝑜𝑔 7 4𝑥+90 − 𝑙𝑜𝑔 7 𝑥=2
𝑥=28
𝑥=9
𝑥=4
𝑥=45.25
𝑥=2