3.4: Properties of Logs
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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?
Log Properties
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 on the left is read " \(b\) to the power \(x\) is \(a\)." The logarithmic equation on the right 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 \(I n\) implies base \(e . I n\) 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 three basic properties of logs that correlate to properties of exponents.
Addition/Multiplication
\(\log _{b} x+\log _{b} y=\log _{b}(x \cdot y)\)
\(b^{w+z}=b^{w} \cdot b^{z}\)
Subtraction/Division
\(\log _{b} x-\log _{b} y=\log _{b}\left(\frac{x}{y}\right)\)
\(b^{w-z}=\frac{b^{w}}{b^{z}}\)
Exponentiation
\(\log _{b}\left(x^{n}\right)=n \cdot \log _{b} x\)
\(\left(b^{w}\right)^{n}=b^{w \cdot n}\)
There are also a few standard results that should be memorized and should serve as baseline reference tools.
- \(\log _{b} 1=0\)
- \(\log _{b} b=1\)
- \(\log _{b}\left(b^{x}\right)=x\)
- \(b^{\log _{b} x}=x\)
Example
Earlier you were asked what \(\log _{10} 1,000\) represents. A log expression represents an exponent. The expression \(\log _{10} 1,000\)
represents the number 3 .
\(\log _{10} 1000=\log _{10} 10^{3}=3\)
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.
Write the expression as a logarithm of a single argument.
\(\log _{2} 12+\log _{4} 6-\log _{2} 24\)
Note that the center expression is of a different base. First change it to base 2 by switching back to exponential form.
\(\begin{aligned} \log _{4} 6 &=x \leftrightarrow 4^{x}=6 \\ 2^{2 x} &=6 \leftrightarrow \log _{2} 6=2 x \\ x &=\frac{1}{2} \log _{2} 6=\log _{2} 6^{\frac{1}{2}} \end{aligned}\)
Thus the expression with the same base is:
\(\begin{aligned} \log _{2} 12+\log _{2} 6^{\frac{1}{2}}-\log _{2} 24 &=\log _{2}\left(\frac{12 \cdot \sqrt{6}}{24}\right) \\ &=\log _{2}\left(\frac{\sqrt{6}}{2}\right) \end{aligned}\)
Prove the following log identity:
\(\log _{a} b=\frac{1}{\log _{b} a}\)
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.
\(\begin{aligned} \log _{a} b &=x \\ a^{x} &=b \\ a &=b^{\frac{1}{x}} \\ \log _{b} a &=\log _{b} b^{\frac{1}{z}}=\frac{1}{x} \\ x &=\frac{1}{\log _{b} a} \end{aligned}\)
Therefore, \(\frac{1}{\log _{b} a}=\log _{a} b\) because both expressions are equal to \(x\).
Rewrite the following expression under a single log.
\(\begin{aligned} \ln e-\ln 4 x+2\left(e^{\ln x} \cdot \ln 5\right) &=\ln \left(\frac{e}{4 x}\right)+2 x \cdot \ln 5 \\ &=\ln \left(\frac{e}{4 x}\right)+\ln \left(5^{2 x}\right) \\ &=\ln \left(\frac{e \cdot 5^{2 x}}{4 x}\right) \end{aligned}\)
True or false:
\(\left(\log _{3} 4 x\right) \cdot\left(\log _{3} 5 y\right)=\log _{3}(4 x+5 y)\)
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.
Review
Decide whether each of the following statements is true or false. Explain.
1. \(\frac{\log x}{\log y}=\log \left(\frac{x}{y}\right)\)
2. \((\log x)^{n}=n \log x\)
3. \(\log x+\log y=\log x y\)
Rewrite each of the following expressions under a single log and simplify.
4. \(\log 4 x+\log (2 x+4)\)
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. \(\log _{4}\left(\frac{2 x^{3}}{5}\right)\)
9. \(\ln \left(\frac{4 x y^{2}}{15}\right)\)
10. \(\log \left(\frac{x^{2}(y z)^{3}}{3}\right)\)
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. \(\log _{\frac{1}{6}} \frac{1}{x}=\log _{b} x\)
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