12 Logarithms
Logarithms, or logs for short, are the same as powers just written in another way.
12.1 Reverse of indices
\(a\) is called the base of the logarithm. When dealing with logs it’s often useful to think of a numerical example to keep the idea straight in your head.
\[ \begin{aligned} 10^3 &= 1000 \\ 3 &= \log_{10} 1000 \end{aligned} \] This is the same fact written in index notation and as a logarithm.
12.2 Rules of logarithms
Just as there are rules when dealing with indices, there are the corresponding rules when dealing with logarithms too.
We can use these rules to manipulate algebraic expressions. For example, let’s write the following as a single logarithm:
\[ \begin{aligned} 3\log_{10} 2 + \log_{10} 5 - \log_{10} 4 &= \log_{10} 2^3 + \log_{10} 5 - \log_{10} 4 \\ &= \log_{10} 8 + \log_{10} 5 - \log_{10} 4 \\ &= \log_{10} (8 \times 5) - \log_{10} 4 \\ &= \log_{10} 40 - \log_{10} 4 \\ &= \log_{10} (\frac{40}{4}) \\ &= \log_{10} (10) \\ &= 1 \end{aligned} \] This is how it was done:
- First we used the power rule \(\log_{a} x^n = n\log_{a} x\),
- then the addition rule \(\log_{a} x + \log_{a} y = \log_{a} xy\),
- and finally, the subtraction rule \(\log_{a} x - \log_{a} y = \log_{a}{\frac{x}{y}}\).
- Then notice \(\log_{10} (10)= 1\) since \(10^1=10\).
Have a go at these simplification questions.
12.3 Solving equations with logarithms in
For example, let’s solve \(3\log_{10} x + \log_{10} 2 = \log_{10} 250\). First we’ll apply the power rule \(\log_{a} x^n = n\log_{a} x\), then the addition rule \(\log_{a} x + \log_{a} y = \log_{a} xy\):
\[ \begin{aligned} 3\log_{10} x + \log_{10} 2 &= \log_{10} 250 \\ \log_{10} x^3 + \log_{10} 2 &= \log_{10} 250 \\ \log_{10} 2x^3 &= \log_{10} 250 \end{aligned} \]
Now since the two sides are equal the values inside the logarithm must be equal. We can then go ahead and solve the resulting equation as normal.
\[ \begin{aligned} \log_{10} 2x^3 &= \log_{10} 250 \\ 2x^3 &= 250 \\ x^3 &= 125 \\ x &= \sqrt[3]{125} \\ &= 5 \end{aligned} \]
Have a go at the following questions:
12.4 Some important bases
Some bases in logarithms come up more than others, because of that some bases have their own notation.
12.4.1 The natural logarithm
A logarithm that has \(e\) as it’s base is known as the natural logarithm and has it’s own symbol.
12.4.2 Base 10
A logarithm that has \(10\) as it’s base has it’s own symbol.
You just don’t bother writing the base.
12.5 Differentiating \(\ln{x}\)
The rule for differentiating \(\ln{x}\) is:
Use that rule to try the following questions.