Common natural logarithms

1. Common Logarithms
*Find common logarithms and exponential values of numbers
*Solve equations using common logarithms
*Solve real-world applications with common logarithmic functions
Because of their frequent use in real-world and applied
problems, logarithms with base 10 are referred to as common
logarithms. Common logs written without an indicated base are
assumed to be base 10.
log
10
x = log x
The standard calculator log button assumes log base 10.

2. The next step is to use this understanding to solve
exponential equations:
Solve for x:
1. 6
3x
= 81
*Use the properties of
logarithms to pull the variable
out of the equation!
2. 5
4x
= 73 3. 6
x - 2
= 4
x

3. Hmm… That last one took a bit of work
Is there any other way
to solve an equation?
Solve for x:
This time by graphing!
2
x-1
= 5
x - 2
You are determining the value of x that makes
both equations true. Graph the two functions
separately and look for an intersection point.

4. Solve: 2
x - 1
= 5
x - 2
That’s our point!
x 2.76
≈

5. Base e and Natural Logarithm
(Natural Logarithms are better for your health)B
*Find natural logarithms of numbers
*Solve equations using natural logarithms
*Solve real-world applications with natural logarithmic
functions
Because e is frequently used as an exponential base,
log base e is defined as the natural logarithm
log
e
x = ln x

6. Natural Log is the inverse of exponential base e = 2.718
Properties of logarithms hold the same for ln x
Here’s the question then:
Why is ln e = 1? Why is e
ln x
= x?

7. Because natural functions and
natural logarithmic functions are
inverses, these two functions can be
used to “undo” each other
xe
x
=
ln
xe
x
=ln

8. Write equivalent Expressions
•Write a equivalent exponential or logarithmic
equation.
• 
• 
5=
x
e
x=5ln
6931.0ln»x
6931.0log»x
e

9. Using e and ln to solve problems and equations:
Solve for x:
•3
2x
= 7
x – 1
2. 6.5 = -16.25ln x
3. 18 = e
3x
5. 10 = 5e
5x