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Properties of Logarithms

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Properties of Logs

Log functions are inverses of exponential functions.  This means the domain of one is the range of the other.  This is extremely helpful when solving an equation and the unknown is in an exponent.  Before solving equations, you must be able to simplify expressions containing logs.  The rules of exponents are applied, but in non-obvious ways.  In order to get a conceptual handle on the properties of logs, it may be helpful to continually ask, what does a log expression represent?  For example, what does \log_{10} 1,000  represent? 

Watch This

http://www.youtube.com/watch?v=SxF44olWTyk James Sousa: Properties of Logarithms

Guidance

Exponential and logarithmic expressions have the same 3 components.  They are each written in a different way so that a different variable is isolated.  The following two equations are equivalent to one another.

b^x=a \leftrightarrow \log_b a=x

The exponential equation is read “ b to the power  x is a .”  The logarithmic equation is read “log base  b of  a is x ”. 

The two most common bases for logs are 10 and e .  At the PreCalculus level log by itself implies log base 10 and ln implies base e . ln is called the natural log.  One important restriction for all log functions is that they must have strictly positive numbers in their arguments.  So, if you press log -2 or log 0 on your calculator, it will give an error. 

There are four basic properties of logs that correlate to properties of exponents. 

Addition/Multiplication:

\log_bx+\log_by=\log_b(x \cdot y)

b^{w+z}=b^w \cdot b^z

Subtraction/Division:

\log_bx-\log_by=\log_b \left(\frac{x}{y}\right)

b^{w-z}=\frac{b^w}{b^z}

Exponentiation:

\log_b(x^n)=n \cdot \log_bx

(b^w)^n=b^{w \cdot n}

There are a few standard results that should be memorized and should serve as baseline reference tools.

  • \log_b 1=0
  • \log_bb=1
  • \log_b(b^x)=x
  • b^{\log_b x}=x

Example A

Simplify the following expressions:

  1. \log_4 64
  2. \log_{\frac{1}{2}} 32
  3. \log_3 3^5
  4. \log_2 128

Solution:

  1. \log_4 64=\log_4 4^3=3 \cdot \log_4 4=3 \cdot 1=3
  2. \log_{\frac{1}{2}} 32 = x \ \text{can be rewritten as} \ \left(\frac{1}{2}\right)^x&=32.\\2^{-x} &= 32\\x &= -5
  3. \log_3 3^5=5 \cdot \log_3 3=5
  4. \log_2 128=\log_2 2^7=7

Example B

Write the expression as a logarithm of a single argument.

\log_2 12+\log_4 6-\log_2 24

Solution:  Note that the center expression is of a different base.  First change it to base 2 by switching back to exponential form.

\log_4 6 &= x \leftrightarrow 4^x=6\\2^{2x} &= 6 \leftrightarrow \log_2 6=2x\\x &= \frac{1}{2} \log_2 6=\log_2 6^{\frac{1}{2}}

Thus the expression with the same base is:

\log_2 12+\log_2 6^{\frac{1}{2}}-\log_2 24 &= \log_2 \left(\frac{12 \cdot \sqrt{6}}{24}\right)\\&= \log_2 \left(\frac{\sqrt{6}}{2}\right)

Example C

Simplify the following expression: 2 \log_{12} 144^{-4} .

Solution:  2 \log_{12} 144^{-4} =-8 \cdot \log_{12} 12^2=-16 \cdot \log_{12} 12=-16

Concept Problem Revisited

A log expression represents an exponent.  The expression \log_{10} 1,000  represents the number 3.  The reason to keep this in mind is that it can solidify the properties of logs.  For example, adding exponents implies bases are multiplied.  Thus adding logs means the bases of the exponents are multiplied. 

Vocabulary

A logarithm is a way of rewriting exponential equations to isolate the exponent. 

Guided Practice

1. Prove the following log identity:

\log_ab=\frac{1}{\log_ba}

2. Rewrite the following expression under a single log.

\ln e-\ln 4x+2 (e^{\ln x} \cdot \ln 5)

3. True or false:

(\log_3 4x) \cdot (\log_3 5y)=\log_3 (4x+5y)

Answers:

1. Start by letting the left side of the equation be equal to x . Then, rewrite in exponential form, manipulate, and rewrite back in logarithmic form until you get the expression from the left side of the equation.

\log_a b &= x\\b^x &= a\\b &= a^{\frac{1}{x}}\\\log_b a&= \frac{1}{x}\\x &= \frac{1}{\log_ba}

Therefore,  \frac{1}{\log_b a}=\log_a b because both expressions are equal to x .

2. \ln e-\ln 4x+2(e^{\ln x} \cdot \ln 5)

&= \ln \left(\frac{e}{4x}\right)+2x \cdot \ln 5\\&= \ln \left(\frac{e}{4x}\right)+\ln (5^{2x})\\&= \ln \left(\frac{e \cdot 5^{2x}}{4x}\right)

3. Note, it may be very tempting to make errors in this practice problem. 

False.  It is true that the log of a product is the sum of logs.  It is not true that the product of logs is the log of a sum. 

Practice

Decide whether each of the following statements are true or false.  Explain.

1. \frac{\log x}{\log y}=\log \left(\frac{x}{y}\right)

2. (\log x)^n=n \log x

3. \log x+\log y=\log xy

Rewrite each of the following expressions under a single log and simplify.

4. \log 4x+\log(2x+4)

5. 5 \log x+\log x

6.  4 \log_2 x+\frac{1}{2} \log_2 9-\log_2 y

7.  6 \log_3 z^2+\frac{1}{4} \log_3 y^8-2 \log_3 z^4 y

Expand the expression as much as possible.

8.  \log_4 \left(\frac{2x^3}{5}\right)

9.  \ln \left(\frac{4xy^2}{15}\right)

10.  \log \left(\frac{x^2 (yz)^3}{3}\right)

Translate from exponential form to logarithmic form.

11. 2^{x+1}+4=14

Translate from logarithmic form to exponential form.

12. \log_2 (x-1)=12

Prove the following properties of logarithms.

13. \log_{b^n}x=\frac{1}{n} \log_b x

14. \log_{b^n}x^n=\log_b x

15.  \log_{\frac{1}{b}}\frac{1}{x}=\log_b x

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