Algebra/Logarithms

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Logarithms (commonly called "logs") are a specific instance of a function being used for everyday use. Logarithms are used commonly to measure earthquakes, distances of stars, economics, and throughout the scientific world.

Logarithms[edit]

In order to understand logs, we need to review exponential equations. Answer the following problems:

1) What is 4 to the power of 3?

2) What is 3 to the power of 4?

3) 2^5 \!

4) 5^2 \!

After you finish, check your answers:

1: "4 to the power of 3" means "4 multiplied 3 times." Thus, 4 to the power of 3 equals 4 times 4 times 4. 4 times 4=16. 16 times 4=64. So 4 to the power of 3 equals 64.

2: 3 times 3 = 9. Times 3 equals 27. Times 3 equals 81. So 3 to the power of 4 equals 81.

3: 2 ^ 5 = 2 * 2 * 2 * 2 * 2 = 32 \!

4: 5 ^ 2 = 5 * 5 = 25 \!

Just like there is a way to say and write "4 to the power of 3" or "4^3 \!, there is a specific way to say and write logarithms.

For example, "4 to the power of 3 equals 64" can be written as: 4^3=64 \!

However, it can also be written as:

 \log_4 (64) = 3 \

Once, you remember that the base of the exponent is the number being raised to a power and that the base of the logarithm is the subscript after the log, the rest falls into place. I like to draw an arrow (either mentally or physically) from the base, to the exponent, to the product when changing from logarithmic form to exponential form. So visually or mentally I would go from 2 to 5 to 32 in the logarithmic example which (once I add the conventions) gives us:  2^5 = 32 \!

So, when you are given a logarithm to solve, just remember how to convert it to an exponential equation. Here are some practice problems, the answers are at the bottom.

Properties of Logarithms[edit]

The following properities derive from the definition of logarithm.

If  b>0 with  b<>1 , then for every real y,a,c it is:

1) \log_b (y^a) = a \log_b (y) \

2) \log_b (b^a) = a  \

3) \log_b (a c) = \log_b (a)+\log_b (c) \

4) \log_b (a/ c) = \log_b (a)-\log_b (c) \


There is also the "change of base rule":

 \log_b (a) = \frac{\log_d (a)}{\log_d (b)} for any  d>0, d\neq1

Proof

Let us take the log to base d of both sides of the equation  b^c = a :

 \log_d (b^c) = \log_d (a) .

Next, notice that the left side of this equation is the same as that in property number 1 above. Let us apply this property:

 c \log_d (b) = \log_d (a)

Isolating c on the left side gives

 c = \frac{\log_d (a)}{\log_d (b)}

Finally, since  c = \log_b (a)

 \log_b (a) = \frac{\log_d (a)}{\log_d (b)}

This rule allows you to evaluate logs to a base other than e or 10 on a calculator. For example,  \log_3 (12) = \frac{\log_{10} (12)}{\log_{10} (3)} = 2.262


  • Solve these logarithms
    •  \log_3 (81) = \!
    •  \log_6 (216) = \!
    •  \log_4 (64) = \!
  • Evaluate with a calculator
    •  \log_4 (6) = \!
  • Find the y value of these logarithms
    •  \log_3 (y) = 3 \!
    •  \log_5 (y) = 4 \!
    •  \log_9 (y) = 4 \!

Answers[edit]

  • Solve these logarithms
    • 4
    • 3
    • 3
  • Evaluate with a calculator
    • 1.29248
  • Find the y value of these logarithms
    • 27
    • 625
    • 6561

Logarithms are the reverse of exponential functions, just as division is the reverse of multiplication, for example:

5 \times 6 = 30 and 30/6 = 5

7^{3} = 343
\log_{7} 343 = 3

Or, in a more general form, if a^b=x, then \log_{a}x=b. Also, if f(x)= a^x, then f^{-1}(x)= \log_{a}x, so if the two equations are graphed each is the reflection of the other over the line y=x. (in both equations, a is considered to be the base).

Because of this, a^{\log_{a}b}=b and \log_{a}a^b=b
.

Common bases used are bases of 10 which is a called a common logarithm or e which is called a natural logarithm" (e~=2.71828182846).

Common logs are written either as \log_{10}x or simply as \log x.

Natural logs are written either as \log_{e}x or simply as \ln x (the ln stands for natural logarithm).

Logarithms are commonly abbreviated as logs.

Properties of Logarithms[edit]

  1. \log_{a}x + \log_{a}y = \log_{a}x*y
  2. \log_{a}x - \log_{a}y = \log_{a}\frac{x}{y}
  3. \log_{a}x^b = b \times \log_{a}x

Proof:
\log_{a}x + \log_{a}y = \log_{a}x*y

\log_{a}x + \log_{a}y

\log_{a}x = b and \log_{a}y = c

\ a^b = x and \ a^c = y

\ xy = a^b a^c

\ xy = a^{(b+c)}

\log_{a}xy = b + c

and replace b and c (as above)

\log_{a}xy = \log_{a}x + \log_{a}y

Change of Base Formula[edit]

\log_{y}x=\frac{\log_{a}x} {\log_{a}y} where a is any positive number, distinct from 1. Generally, a is either 10 (for common logs) or e (for natural logs).

Proof:
\log_{y}x = b

\ y^b = x

Put both sides to \log_{a}

\log_{a}y^b = \log_{a}x

\ b\log_{a}y = \log_{a}x

\ b = \frac{\log_{a}x}{\log_{a}y}

Replace \ b from first line

\log_{y}x = \frac{\log_{a}x}{\log_{a}y}

Swap of Base and Exponent Formula[edit]

a^{\log_{b}c}=c^{\log_{b}a} where a or c must not be equal to 1.

Proof:

 log_{a}b = \frac{1}{log_{b}a} by the change of base formula above.

Note that a=c^{log_{c}a}. Then

a^{log_{b}c} can be rewritten as

({c^{log_{c}a}})^{ log_{b}c} or by the exponential rule as

c^{{log_{c}a}*{log_{b}c}}

using the inverse rule noted above, this is equal to

c^{ {log_{c}a} * { \frac{1}{log_{c}b} } }

and by the change of base formula

c^{log_{b}a}